Aaagh! I've just spent about half an hour carefully researching an answer, and typing out all of the info, only to have gone and clicked something that's deleted most of what I'd typed! Grrr!
Here's a summary of what I was trying to submit:
Realistically, you'll probably need to travel out on the Friday evening and back on the Sunday, hopefully finding somewhere convenient to stay in Exeter for the 2 nights (unless, of course, you're staying with friends or relatives anyway).
There are direct trains from Basingstoke to Exeter St David's at 07 minutes past each hour on the Friday evening, taking about 2 hours 37 minutes to get there. The off-peak return fare is surprisingly low, at just £48 but you can get that down to under £40 by purchasing two Advance singles. (Deduct 34% from those figures if you've got a Railcard).
If you've not got accommodation in Exeter, I suggest staying at the St Andrew's Hotel, as sites such as Booking.com are currently offering your 2-night stay for a total of £90:
https://www.st-andrews-hotel.com/
I've stayed there myself and I'm happyto recommend it to you. It's possible to walk there from Exeter St David's but it's a lot easier to take a train from there to Exeter St Thomas (which you can do without paying another fare, as your ticket will be valid to any Exeter station), from where it's a much shorter walk.
Travelling back on the Sunday, direct trains depart at 24 or 25 minutes past each hour from Exeter St David's but it might be best to avoid morning departures, in case there are any knock-on effects from the previous day's strike action.