Base ISO can be thought as "unamplified sensitivity of the camera." Amplification of a signal always leads to image quality degradation, most redly seen in lower signal to noise ratios (SNR), but also in a camera's decreased dynamic range (DR) and color sensitivity. The basic premise of testing the camera's base ISO is therefore to check at which ISO the image quality parameters is the highest (SNR, DR and color sensitivity). This can be done in a variety of ways, here are two:
1) through specialized tests that use a test image and controlled conditions (lighting, exposure) for comparison and later processes the result in specialized software. Although possible to do by yourself given enough effort, probably "checking the website" saves you a lot of hassle, for example the DXO mark website posts these kinds of ISO curves.
2) "by eye at home" If you want to check how ISO is affecting your images in real life, you want to provide the perfect conditions for noise or decreased image quality in the form of lost detail to manifest itself (because DR may be difficult to judge by eye at low ISO's). These conditions are:
Shoot RAW with no noise reduction,
shoot a scene with a mixture of black elements (for bringing out shadow noise)
Include detailed red textures.
for CCD Usually percieved detail is lost quickest in red textures because of the lower number of blue and red photo-sites compared to green in a typical CCD array coupled with our eye's large sensitivity to changes in the red channel.
for Foveon sensors Color details are lost for these sensors through a slightly different mechanism. In these sensors the topmost layer ("blue") has the largest amount of photo-sites while the middle ("green") and bottommost ("red") have fewer (for instance: the Quattro technology has 20 MP for topmost, but only 4.9 for the middle and bottom layers), This is coupled by the fact that most of the light gets filtered through the first two layers. At lower light intensities, this gives rise to very few photons reaching the lower layer, and thus false colors and color "cross-talk" in the shadows, especially false red values. I suspect that red details that differ slightly in hue are lost quite quickly in the shadows, but less so in the mid tones and highlights, but I cannot confirm this as I don't have such a camera. Fortunately for our tests, shadow noise and loss of image quality is very noticeable in this type of sensor so this is not a huge factor in evaluating the base ISO.
Shoot a series of suspected base ISO values (ex. 50, 100, 200, 400) keeping the lighting and exposure constant, and judge them side by side. To judge the noise performance, lighten the blacks to get an equal exposure and magnify that area. To judge the lost detail, magnify the red textured area (you can bump up the contrast to make it more visible, but be careful to do it equally on all images).
So what if the images are identical? Sometimes both of the lowest ISO images will be identical in RAW. This can be due to two factors, 1) the difference in signal to noise ratios and detail loss is not discernible to our eyes between the ISO's (we need to use a more rigorous method of testing) 2) the lower ISO setting basically takes the same RAW image but later pulls it down one stop (in this case the higher ISO setting is actually the base ISO).
Disclamer If you are shooting for the highest possible IQ, always shoot at ISO's that are multiples of two of your base ISO, this is due to the fact that only at those ISO there is a hardware implementation of amplification, at all other values, the camera just digitally pushes the exposure (this is why for example SNR can be higher at ISO 700 than 800). This is only up to a certain point (usually around 1600 for modern DSLR's) and sometimes not implemented at all for example in cheap point-and-shoots, and does not work for half your native ISO. For more details see: How is ISO implemented in digital cameras?
for a related question see for example: What exactly is "base ISO" and how do I find what is base ISO on my camera?