Is it safe to connect my old flash to my DSLR D3300 using a Hot shoe sync adaptor. Will the AS-15 Sync Terminal Adapter act as an isolator against any high voltages that can affect my camera?
You always need to consider the trigger voltage actually supplied by the flash and the voltage the camera's hot shoe can tolerate without being damaged. Older flashes often have higher voltages than newer DSLRs can tolerate. Since you haven't informed us exactly what kind of "old flash" you are considering, we have no way of finding out what the trigger voltage is for the flash in question.
When the camera's shutter is released by pressing the shutter button all the way down the shutter begins to open. When the first shutter curtain is fully open exposing the entire sensor at the same moment (if the shutter speed selected is slower than the camera's flash sync speed), the connection between the center pin and the ground (on either the hot shoe and/or, with some cameras, the PC terminal connector) is completed inside the camera. When this circuit closes it allows voltage from the flash to flow through the camera and back to the flash where it causes the flash to discharge the energy stored in the flash's capacitor(s).
A word of caution about using an older or undocumented flash with your expensive DSLR: When a charged flash is connected to a hot shoe and the hot shoe completes the circuit a lot of voltage can flow through the camera's circuitry. If the voltage supplied by the flash is higher than the camera can tolerate, you will probably irreparably fry your camera's circuitry, or at least those parts connected to the hot shoe.
Different cameras have different flash voltages they are designed to tolerate without damage. Some can only handle a few volts in the 6-10 volt range. Others can work with flash output voltages of 250 volts or more. Check the specifications for your camera and do not allow more voltage than for which your camera is rated to flow through the hot shoe! Most Nikon cameras can tolerate 250V, but you need to confirm this is the case for your D3300. What does the user manual say?
There are voltage reducers available, but even those can be overloaded beyond their specifications. The one linked above is rated to reduce up to 400 volts to less than 6 volts.
The Nikon AS-15 Sync Terminal Adapter provides no voltage protection and is not recommended for flashes that supply more than 6V.
Another safer option is to use a set of cheap manual wireless triggers. At the worst case scenario you'll only fry the receiver of an $18 set instead of your camera.
Will the AS-15 Sync Terminal Adapter act as an isolator against any high voltages that can affect my camera?
No, it will not.
B&H has the following note on their listing for the Nikon AS-15:
Note: This does not provide high-voltage sync protection, and is not recommended for flash units that have more than 6v.
See also: What is a Sync Terminal Adapter?
What you're looking for is a Wein SafeSync or other voltage limiter.
The question here is what the sync voltage on your old flash is. The Nikon D3300 can take 0V-250V on the hotshoe, according to its manual (i.e., negative voltages will also fry the hotshoe). So, as long as your flash's sync voltage is in that range, it won't fry your camera's hotshoe.
The botzilla page has notes on how to measure sync voltage, if you have a voltmeter.