Chosen Solution
Getting photos off 2006 Mac Pro without original monitor. Which patch cord to buy between Mac and new MacBook Air. Trying to use the MacBook Air as a monitor
I don’t think the MacBook air will work as a monitor for that machine. Maybe look up guides on how to connect it to an external monitor like a TV or something.
Do you know if the drives are encrypted? If not, get a 3.5” enclosure and pull them out of the Mac. If they are, then you may not be able to do it, or it’ll ask for the FileVault password (usually your user account password)/recovery key. Once you can get into the drive, open the Users folder and then find the account. However, you may need to use chmod which you can read about here. Chmod IS NOT FOR THE FEINT OF HEART, and needs to be restored back for proper security once you’re finished if you put the Mac Pro back into production. The other option is simply to buy a new monitor, or if you do not use it all the time pick up a descent one with HDMI from a thrift store that works to save a few bucks - hopefully at least FHD (1080p). DVI can work, but it’s usually a sign of a really old monitor that’s commonly CCFL which are all so old they’re dying left and right these days. HDMI can be adapted to DVI with a DVI>HDMI cable for $8, so stick to HDMI and monitors with LED backlighting.
There is one method that can allow you to leverage the MacBook Air as a monitor to the Mac Pro. Here we will need a Thunderbolt cable to connect the two systems so you can leverage Target Disk Mode. Basically, your converting the Mac Pro into a glorified external HD system and using the MacBook Air access of the drives. You’ll still need another drive to copy the files to as the internal drive will likely be to small for all of the files. Reference: Understanding the Applications for Target Disk Mode