Basil plants that are that far along are, in my opinion, too mature to split. While three in a pot will never do as well as a single plant in a pot, I think at this point you may be better off just to leave them, pinching each back with some regularity. Transplanting certainly would be a shock, if not fatal.

If you really want to try, this is how I would do it: cut each back by 1/3 (cutting just above a set of leaf nodes) and then knock the clump out of the pot. Immerse the root ball in a bucket of water and very gently, slowly, and patiently swish around and tease apart the intertwined roots of the 3 plants as much as possible in order to separate them.

Rather than put such large plants into the ground (soil being heavy and the roots already struggling to survive such treatment) I'd opt for 3 individual pots of light soilless mix (NOT potting soil!!). The roots of each plant will resist being spread apart while so wet, but do your best to do so when potting them individually.
Once potted, as with anything, firm the mix gently around the roots and water them in with seaweed solution to help minimize transplant shock and supply some trace minerals.

Leave them in a protected, semi-shaded site for 4-5 days before putting them into full sun. Remember that potted plants need regular (daily) watering when in full sun, esp. if in dark-colored pots.

Good luck!

Thank you, weedlady, for the detailed response. Since I have two pots, I think I'll try it on one of them, probably the one with three plants, teasing apart gently according to your instructions. Actually, having read them, I'm sort of keen to try it just for the sake of the experiment.

The reason I posted is that in the past I've thought nothing of pulling (usually smaller) basil plants apart when several came in the pot. I actually sought out pots with several plants, thinking I'd made out. But the results were never satisfactory; in fact, most times the plants never fully recovered and more or less sat there, sulking and looking back at me for the rest of the season. This ran counter to my practice with, say, petunias and other annuals that seemed to recover from just about any abuse I meted out to them, and quickly. Basil seemed different--something you knew and that I now know. Again, thanks for taking the time. Gary

You can tease them apart, but you should pinch them bask sharply because the current foliage will put too much of a strain on the damaged roots. Once the roots settle themselves, the plants will set new leaves. Leave them out of direct sun and avoid fertilizing after transplanting for the same reason.

Good luck.