Issues programming Arduino Uno with Sparkfun's MAX3232 breakout board
Switching the power off to the MAX3232 appears to be okay.
There are no components connected to the Rx and Tx lines on the breakout board to complicate it.

Unfortunately the break out does not give access to pin 1, EN which is "Enable Control", or pin 18 SHDN "Shutdown Control". AFAICT they could have been used to turn off the receiver and transmitters of the MAX3232. That would have been ideal. Maybe contact Sparkfun and suggest the improvement to put them on pads for folks to use, and offer to beta test it :-)

The datasheet for the MAX3232 says that it is able to withstand a continuous short on the Tx line, so it sounds pretty tolerant to minor abuse. Also, the diagrams dont show any pull-ups on the MOS/TTL Tx line. So I would expect it t be okay if you shut the power off, but continued to wiggle the Tx Rx pins.

An alternative would be to disconnect the Tx pin from the Arduino, so that the MAX3232 wouldn't interfere with the host PC's USB comms. It sounds like the thing on the RS232 connection will ignore data it receive while the Arduino is being programmed. If that is not true, then it would need to be disconnected too.

At the moment, switching off power looks like a reasonable approach.
Disconnecting Tx and Rx also looks reasonable, and slightly better than powering the MAX3232 off.

Do you know how you want to do that?

NB: The MAX232 datasheet shows that part should not be driven by a high when the chip is unpowered. However the MAX3232 datasheet does not carry that warning. My preference would be to use the MAX3232 signals 'EN' and 'SHDN'. However those are not available on the breakout. I'd recommend having the part powered, and disconnecting Tx and Rx from the Arduino.