It's generally easy to find an affordable apartment in Basel, provided you have a realistic budget. It's not even close to the awful situation in Zurich, where the only somewhat acceptable places are luxury apartments the average person cannot possibly afford.
That being said, I'm really sorry, but 1350 for two people is not a realistic budget for any city in this country. Not for an apartment big enough for two people anyway, unless you're fine with living in a somewhat older 2/2.5-room apartment (i.e. 1 living room, 1 bedroom) in a mediocre neighborhood (though needless to say, in Switzerland, even mediocre is still good enough). Basel might be cheaper than Zurich or Geneva, but it's not THAT cheap. If you want a somewhat modern 2/2.5-room apartment, you'll pay 1500-2000 in Basel too. If you want a 3-4-room apartment, expect to pay at least 1800, unless you really happen to stumble upon an awesome deal through someone you know, but that's hardly a scenario you can count on...
So it can either be a small apartment in a city - or you need to move to the suburbs, even though whether this actually makes a difference from a financial standpoint much depends on where you move exactly.
Also, I don't really get why you want to move to Basel, of all places, if you don't know the city at all and don't work here? Where/how do you live right now, since you say you can't find an apartment? I understand your husband is out of work, how about you though? Where do you work? You do realize Basel is some 1.5-2 hours by public transportation from Winterthur, right? Or are you planning to move because you're changing jobs or your husband is hoping to find a new one here? If so, then I would really strongly recommend to look for a job first and then move, not the other way around.
If there's no real reason to move here, then there's thousands of options in the Zurich/Aargau area. What is it you didn't like about Winterthur? I used to live there for a couple of years and live in Basel now, I can tell you it does have some similarities (in my opinion, others might disagree). So depending on what it is you didn't like, you won't like Basel either.
Last but not least, I assume you're rather tight on budget generally, so please be aware that health insurance and taxes are considerably more expensive in Basel than in Winterthur, or really most places in the Zurich area. Health insurance probably for both Swiss and non-Swiss, tax for sure for Swiss, not sure about non-Swiss on whatever permit.