After gathering more information about your bass through comments I am going to offer some thoughts.
It is clear from the description of the problem that for whatever reason, you are getting fret buzz. Fret buzz most commonly results from the vibrating string contacting another fret. Most often this is because the string is too close to the fret it is contacting due to factors discussed below. More rare is a fret that is too high that one or more strings may contact when played.
It is common to get more buzz on the lower strings because they have a wider oscillation arc. And you mentioned no buzz on open strings which eliminates most other buzz producing culprits (such as loose hardware).
Several things can cause fret buzz. Most common is the action (amount of relief between the strings and fretboard) is too low. But there are several ways to adjust the action.
On your bass, you can adjust the truss rod to provide more relief. In most cases a slight bow in the center produces the best results. Completely straight neck or backbow (curved up in the middle) will produce fret buzz in most cases.
But there are other adjustments that can impact the action as well. If I am not mistaken, your AriaPro II FB has a bolt on neck. The neck angle can be adjusted (by a guitar tech or luthier). If your neck is angled back too much, it can contribute to fret buzz as well due to lowering the height of strings above fretboard (action). Be sure your neck is set at the the optimal angle.
You can also change the action by raising or lowering the bridge. After confirming that the neck angle is correct and adjusting the truss rod to provide a slight bow to counteract the string tension wanting to pull the neck back straight, you can adjust the bridge height to fine tune the action to get the best compromise between playing comfort and buzz free performance.
Since your instrument is quite "vintage" it is possible that the slots in the nut have become worn and the strings could be sitting too close to the frets on the headstock end of the instrument as opposed to the bridge. If the strings get way too close to the frets at the nut end - you might even get some buzzing on your lowest strings when played open. If this is contributing to fret buzz, you can have the nut replaced and properly slotted for the gauge strings you use.
Another thing that can cause fret buzz is worn frets. Over time - the strings will wear grooves in the frets and wear them down until so the strings are not as high as they need to be when fretted. Closely examine your frets (particular under the strings) to see if there are any deep grooves or if they appear to be excessively worn. Eventually, if played often enough, almost every guitar or bass that uses metal strings will require a "fret job" or re-fret.
Since there are so many different things that might need adjustment on your bass, I would encourage you to take it to a qualified guitar technician for a full set up and adjustment. I am sure that whatever is causing your fret buzz (and it may be a combination of factors) it can be corrected and your instrument will be fully capable of providing you a very satisfying playing experience. Good luck!