2 or 3 years ago it used to be hard to find but now a days it is pretty much everywhere. EG. all Manor, Coop and Migros restaurants have it, Starbucks, McDonalds, Hitzbergers, Movenpick cafe in Marktplatz, many SBB stations (including Basel) have a wireless hotspot, not just the number 10 trams but also the 17 and a couple of the number 8's have it, as do many postbus services (although these are more in BL than BS). And more unusual service providers have it - my rather staid (but good
) hairdresser in Aschenplatz surprised me a few weeks ago by providing a printed out slip with a password for 2 hours free wifi in response to my query, as did my dentist, just yesterday.
So my advice is, just go where you normally go and do what you normally do but ask if they have wifi ... as often they don't advertise the fact. I think you will be pleasantly surprised.