There is no GFCI outlet in my basement, and circuit breaker reset won't work
If the entire circuit is on one breaker, the problem is not likely to be at the panel. I can imagine two possibilities: 


A connection at an outlet is loose and failing. Take a guess at which outlet is the last working one upstream of the problem. With the breaker switched off, open the box and check the connections. 
Damage to the wiring has occurred due to rodents, impact, corrosion, etc. This might be more difficult to diagnose if the walls are finished. You might use continuity testing to check the individual conductors. Again, switch off the breaker first. 


A third less likely possibility is that there are two "home runs" connected at the breaker, and one has come loose. If that was the case, however, you'd probably see intermittent failure of the rest of the circuit as well.