The base coaches are restricted to the area of the coaches box up until the release of the pitch (and, by extension, between pitches beginning when the pitcher is in position and preparing to pitch).
After a pitch is released, the base coaches are free to roam around a bit, which they might do to direct runners, avoid a fielder, avoid the ball, etc. But wherever they move, they still are obligated to not interfere with a play!
SoftballDad, if the umpire in your game is on the ball, you shouldn't have an issue with a base coach interfering and preventing a catch. Because if he did, then the batter would be out anyway!
In the case of a fair or foul fly ball, if the coach interferes and prevents a catch, the batter is out. If the coach interferes with any other type of play, then the ball is dead and the runner closest to home is out.
On any batted ball where the fielder has an opportunity to make a play (ie: record an out) the base coach must vacate any space or area required by the fielder to make the play. And the box itself is not a sanctuary against an interference call. Even if the ball is coming down right in the middle of the box, the coach MUST vacate that area. In essence, when making a play the fielder has 100% right of way and the coach has 100% obligation to not interfere (either on purpose or by accident).
The above applies when the coach interferes with a fielder (the actual defensive player). When a coach contacts a batted or thrown ball, the rule is a little bit different. This is only interference if the coach intentionally contacts the ball. If a coach is accidently hit by a batted or thrown ball, there is no rule infraction and play continues.