Basel offers many affordable accommodation possibilities. If you are flexible with transportation, you can also check options in Germany (along tram no. 8 to Weil am Rhein) or France (along tram no. 3 to St. Louis). You may find the best solution yourself by using some platforms like booking.com or airbnb.com 

Some reccommendations, all in walking distance of max. 15 minutes:

Hotel becozy (self checking, no breakfast, min. stay 2 nights, affordable price)

Hotel Dorint (single room CHF 134, double CHF 154, send an email and mention the code 4495)

Hostel Silo (special architecture, low price)