Some cameras may have the current 'mode' identified in the web interface (assuming IP camera). It will say 'Day' or 'Night' or 'Open' or 'Closed' to indicate if it's in day or night mode.
Unfortunately you can't rely on it 100% as the selection may change to the correct value since the proper command was sent for the day/night switch. But if the filter is physically stuck and will not move even if the camera 'thinks' it did.
A more full proof method, if you have a camera with a removable lens you can unscrew the lens and look at the imager itself. If the glass is clear, with no tinting, then the filter is away from the sensor and the camera is in night mode. If the glass has a tint to it (Blue, Green, Red, etc) then it is in day mode. You can do this at night with a flashlight to know for sure which mode the camera is in.
Backing up to answer the original question, if you're not using IR at night the difference between performance with the cut filter on and off is not going to be the difference between a camera sucking and not sucking.
It's more like it sucks at 7.5 level when it should only suck a 7 on the suck scale (to put it in technical terms). Ambient IR generally is not enough to affect performance in a drastic fashion.