Sure, you can "up-rez" and SD... but the end results aren't guaranteed.
What kind of old tapes are we talking about here? VHS? S-VHS? Hi8? Betamax? U-Matic? BetaSP? DigiBeta? The better the quality of the existing recording, the better results you'll get. That said, you also need a very high-quality interface to capture to your computer, and the hardware/software combination will need to be able to do one of two things: either capture uncompressed 4:4:4 SD to be converted to HD later, or to up-rez on the fly to HD specs.
And of course, you then have to work with the 4:3 frame in a 16:9 HD spec, which usually requires pillar box to keep it from looking stretched.
Now, for the easy way out (which isn't cheap), you could take it to a post house and let them do it. Not the cheapest, but probably the most reliable.