Yes, there are multiple flights from Qatar to Basel for under 3,609﷼. The cheapest flight booked recently is on Turkish Airlines for just 3,436﷼, but on average you can expect to pay 4,422﷼.
There are no direct flights from Qatar to Basel, but Cheapflights can provide you with all the information you need to find the fastest route with the fewest stops.
For as low as 3,440﷼, Turkish Airlines can have you in Basel from Qatar in 9h 10m. This flight may include one or multiple stops.
Flying on Turkish Airlines is currently your cheapest option to Basel, with prices starting at 3,437﷼. Another option worth looking into is British Airways, starting at 6,389﷼ return.