To blow or not to blow on the joint, that is the question
It depends on what and how you're soldering.  If you are manually soldering things that come off a board, like a wire, then it's usually good to gently blow on the joint to cool it quickly.  The advantages are:

It hardens the joint quickly.  This reduces the time you have to hold things still, which in turn reduces the chance that things are wiggling around as the solder hardens.  When you're holding one or more things with your hands, this is useful.

You can get a visual indication of the quality of the joint.  Seeing the solder harden shows you a bit about how well the solder made thermal contact, which depends on how well it flowed around all the metal parts of the joint.  It's hard to explain without showing it, but sometimes you can catch a problem by noticing that the solder just didn't look right after it cooled, or in the process of cooling.



However, when things are held in place on their own, then you get advantage from letting the solder cool slowly.  This is the case, for example, with soldering a single pin of a larger component on a board.  If the other pins have already been soldered, then they will hold the part in place.  The part joint won't be weakend due to the part wobbling as the solder was cooling.  Now letting the mechanical stresses due to uneven heating dissipate a bit by slowing the hardening process helps.

Of course when you're doing reflow soldering, follow the recommended temperature profile.  In that case the equipment handles that outright, and you shouldn't be in there altering the process.

With manual hot air soldering, you generally don't want to blow on joints either.  You wouldn't use hot air to solder two things you have hold still yourself.  Generally you're heating a entire part.  Hot air heats the board and other parts around where you are soldering.  Give them a chance to cool down slowly and with the least thermal stress.