For audio cassettes, all you really need is a tape deck with a line-out jack (and cord), which you plug into the computer’s line-in, and we record with Audacity (free and open source).
For video, you can get a cheap (<$15) cord/device on eBay that lets you do basically the same thing from a vcr. (It gives you a USB to plug into the computer.) I can’t remember the name of the device we purchased, but if you search AV to USB you should find some options. Of course, if you don’t have a lot of VHS, it may be cheaper (especially when you factor in the time) to just have them done from any of the numerous companies that handle that. Alternately, you can connect VCRs to some camcorders, and then connect to the computer and do them that way.
Oh course all of this has to happen in real time, so it can take a long time. I like to do them on days when not all my staff are in the office, and use a computer that’s otherwise not in use.