I'm new to photography, I hope I won't misuse any terms here.
Even if everything is still and constant will this shutter speed affect image sharpness?
In other words, if I would be to increase the light source and increase shutter speed accordingly to get the same exposure, could my resulting image be sharper?
In theory increasing the shutter speed (in other words, shortening the exposure) won't make the image sharper since there is no motion to create a blur. On the other hand on long exposures the sensor can be a bit more noisy, and at some shutter speeds (around 1/10s) mirror slap can be a problem if you use a DSLR (but most DSLRs have some form of mirror lockup or two-step exposure to mitigate this).
On a tripod indoors, long shutter speeds should not be problematic. If they are, the problem lies elsewhere: poor tripod stability, poor shutter release technique, ignoring dark noise compensation settings in the camera, etc.
A sturdy tripod head, a remote shutter trigger, and knowledge of the camera settings are the way to add long exposure to your photography toolbox.
There are several factors related to the scenario in your question that can be at play that affect image sharpness.
Vibration Effects
Depending upon what kind of camera and tripod one is using, there can be several sources of vibration that can affect the clarity of an image:
Many DSLRs offer mirror lockup to eliminate the effects of "mirror slap". This is often most severe at around 1/15-1/30 second exposure times. Exposures shorter than about 1/100 seconds are usually over before the vibration can influence the result, though I find mirror lockup is helpful for up to 1/250 when using very long focal lengths for imaging the Moon. Exposures longer than about 1 second are long enough that the vibration usually has little influence over the total length of exposure.
Recall that for a thirty second exposure, a person can spend 2-3 seconds walking across the scene and not show up in the image. Do note, though, that if there are bright point light sources in an otherwise dark scene, even short duration vibration can cause "squiggles" from those light sources to appear in an exposure much longer than the duration of the vibration. So if your person is wearing tennis shoes that light up with each heel strike, the lights from the tennis shoes will show up in your 30-second exposure but the rest of the shoes and the person wearing them won't!
Most current MILCs and DSLRs when used in Live View offer various shutter modes that eliminate the movement of a mechanical shutter curtain at the beginning of an exposure. These may be called "electronic first curtain" mode, or sometimes are called "silent shooting modes".
Wired cable releases or wireless shutter remotes that use infrared or radio to actuate the camera's shutter can eliminate motion blur caused by pressing the camera's shutter button, even when the camera is mounted on a tripod.
A heavier camera mounted on a very robust tripod will suffer less from the vibrations caused by mirror slap, shutter movement, and pressing the shutter button on the camera than a lighter camera mounted on a flimsy tripod that can often resonate with the various sources of vibration. The sturdier setup will dampen vibrations much faster while the flimsier setup can actually make them worse!
The specifics of a particular camera, lens, tripod, and shutter release method must be taken into account to determine what is optimal for a particular shot.
In the case of adding light to reduce a three-second exposure to, say, 1/15 second, you may actually increase blurriness due to mechanical vibrations!
Lens Aperture
Lenses almost always perform better at some aperture settings than others. For most conventional designs that have been around for a while (but which are still used in many newer "budget" lenses), the "sweet spot" is usually somewhere around 1-3 stops narrower than the maximum available aperture setting (lowest f-number). An f/2 lens will tend to be sharpest somewhere in the range of f/2.8 to f/4 to f/5.6. An f/5.6 lens often is sharpest at around f/8 to f/11. A few more recent lens designs, particularly on the higher end, are optimized to be sharper closer to the widest available aperture. Many of Canon's Super Telephoto series of "Big White" lenses are just as sharp wide open or stopped down only one-third stop or so as they are at any other aperture setting.
The increasing detail rendered by stopping down a lens is offset by the increasing influence of diffraction as the aperture opening narrows. With digital cameras, the size of the sensor's photosites (a/k/a sensels or pixel wells) affects when the effects of diffraction first become measurable. We call this the diffraction limited aperture (DLA) for that particular camera. For larger sensors with fewer and larger photosites, the DLA can be as high as f/13.2 for the 12.8MP FF Canon EOS 5D "Classic". On the other side of the coin, high density sensors have much lower DLAs. The Canon EOS 90D, with a 32 MP APS-C sensor (that has the same pixel density as an 82 MP FF sensor would), has a DLA of f/5.2.
One can clearly see that with some high density sensors, the DLA begins having influence before the lens is stopped down to its "sweet spot" aperture!
How the aperture setting affects your image taken of a static scene with a tripod mounted camera must take into consideration the specifics of your camera's sensor and its DLA, as well as the specifics of the lens you are using and its "sweet spot".
Signal-to-Noise Ratio
In low light situations, increasing the amount of light actually allowed into the camera almost always increases image detail. This is due partly to the nature of light itself, which has random distribution of photons within a light field of given intensity. We call this randomness "Poisson distribution" noise or "shot" noise.
The less signal an image has, the lower the signal-to-noise ratio (SNR or S/N ratio) will be because some noise is fixed pattern noise that remains essentially the same, regardless of how much signal is present. "Shot" noise is variable, but only increases as the square root of the increase in total light intensity. This means that "shot" noise has the most influence to the SNR at lower light levels and the least influence at higher light levels.
Images with a low SNR tend to have more noise reduction applied by the camera or by most raw conversion applications. Noise reduction tends to reduce detail in an image. So the higher the S/N ratio is, the less detail destroying noise reduction needs to be applied to an image.
Thermal considerations also come into play regarding image noise. The longer a sensor is energized, the warmer it becomes. The warmer a sensor becomes, the more thermal noise it generates. This was more of a problem a few years ago, but can still be a consideration with many current cameras, especially those with sensors capable of very high frame rates when shooting video footage.
Long exposures of a fairly short duration, such as a few seconds, are one area where DSLRs still have an advantage over MILCs. With MILCs, the sensor must be constantly energized and reading out every fraction of a second to provide an image to the camera's electronic viewfinder (EVF) or rear LCD screen. With DLSRs the sensor need only be energized from a fraction of a second before exposure begins until the sensor has been read out when using the optical viewfinder (OVF). If using a DSLR in Live View, it's pretty much the same as using a MILC.
For very long exposures some cameras offer dark frame subtraction. Some manufacturers call it Long Exposure Noise Reduction (LENR). It's most effective for exposures longer than around 30 seconds, but can be enabled on most cameras that offer it for any exposure one second or longer. There are some cameras that force it without giving the user an option to disable it on exposures longer than a specific amount. After the image is exposed another frame is taken with the same settings (ISO and Tv) while the shutter remains closed. The noise generated by this second exposure is subtracted from the first exposure.
Again, one must consider the specifics of the gear one is using to determine how to best utilize that gear for a specific shot scenario.
But if one is decreasing the exposure time to offset the increase in light present in the scene, then most of this benefit is lost. In such a case you're still only allowing the same amount of light into the camera over a shorter exposure time.
In Summary
Without more specifics, it's impossible to answer your question completely.
Adding the total amount of light the camera is allowed to collect always increases image quality until highlights become fully saturated and begin to clip. If one can maximize exposure to be just below that threshold while also using various methods to reduce the influence of vibration, as well as using the lens at its "sweet spot" aperture without exceeding the camera sensor's DLA, then one can expect to get the best results possible.
