There are two ways of curing spill or crosstalk. One is not to let it happen in the first place, the other is somehow to get rid of it once it has occurred.
Centering my comments on spill, as mentioned earlier one way to reduce spill is to record in a non-reflective environment, and use acoustic screens liberally. This cuts down the potential spill paths and the result will be a relatively clean recording. One can also pay close attention to the selection of microphone polar pattern and aiming. Another way is to record each instrument, as much as possible, separately so there is no source of spill, other than perhaps over-loud headphones.
Once spill has been recorded however, then the obvious way to reduce its effect would be to gate it out, or at least to attempt to do so. This technique has been used greatly to advantage on drum kits, where the spill between mics is almost as loud as the signal itself, since the late 1970s when gates first started to become commonly available. The noise gate is however a brute force technique which requires some refinement for best results.
Gates are commonly thought of as being open or closed, on or off. But nearly all commercial noise gates offer a control that determines the degree of attenuation when closed. Set this to maximum and you will certainly hear what the noise gate can do. The interesting thing is however how low you can set this control and still get very useful results. Settings as low as 6dB can be very worthwhile as spill can be cleaned up during periods when there is no signal present, yet the opening and closing of the gate doesn’t draw attention. Expanders are useful too.
Interestingly in the 1980s, Dolby A units were quite often used in decode mode only (no encoding) as sophisticated expanders which split the signal into four separate frequency bands. One slight drawback was that the HF band was processed more heavily than the rest resulting in a dulling of the sound, but that could be compensated by EQ to a certain extent to give a very clean drum sound, probably better than a simple gate, or non-split band expander.
Dolby A units are however mostly in hibernation now so this technique is no longer widely available.