We're doing a pretty weird, high-volume method because we have hundreds of thousands of hours of tapes to do. That said, I think the simplest way to digitize personal Betas is probably to get a USB encoder made by El Gato or Hauppauge or such (one that supports composite inputs, since that's what your VCR will be outputting). We did some work using an El Gato Video Capture USB encoder (https://www.amazon.com/Elgato-Video-Capture-Digitize-iPad/dp/B0029U2YSA ) and it seemed to be very easy to use and produced good results. I think there are a lot more cheap options on the PC side or from other companies, but I don't know as much about those.
That's a 3.5mm (1/8") audio jack. You can use something like this to adapt it, but you will only get audio on the left (white) connector. You would need something like this to split it to both channels.
Old Sony Betamax VCRs from that era use the 3.5mm audio jack instead of the RCA/phone style connector, like VHS and later Betamax VCRs.
I have a new cord and a part of it didn't fit. here is the Amazon link. the base (if that's what it's called) just sticks out.
Sidenote, when there isn't any tapes in the VTR, is it normal to hear static?