If things work OK in "programA" but not in "programB" (B being Premiere) that is usually a codec issue
AVI is simply a wrapper... what is the codec?
The first question at the end of this link has a link to the free GSPOT to find the codec
First, work through all of the steps here http://ppro.wikia.com/wiki/Troubleshooting
If your problem isn't fixed after you follow all of the steps, report back with the DETAILS asked for in the questions at the end of that link
Could possibly be a sampling rate mismatch. NLE's usually require 48 kHz but can be happy with 44.1 kHz.
Some cameras can create 4-channel, 32 kHz audio which makes many NLE's very unhappy...
Who did the original transfers and what were the settings...??
Redigitize the Beta tapes at 48 kHz, 16-bit stereo.
Is the waveform clipped? We have had issues capturing from our Beta deck where the audio leaving the deck is overmodulated and unfortunately, there is no output level adjustment on the deck. To capture, we have had to go into the sounds control panel in Windows and slide the appropriate level handle nearly al the way to the bottom. It may be that it was captured to DV Cam tape too hot.
Check for clipping.