The correct thing to do is grab yourself some RG-59, or (even better, but hard to work with) RG-6. This is 75-ohm transmission line and is what VGA is based around. This is readily available for cable television usage.
Terminate the signal lines into the RGB lines on the HD-15, and terminate the shield to the corresponding "shield return" pin directly below them on the middle line. You can use normal wire for sync, but if you've got the coax, it doesn't hurt anything to use that. You will likely have to solder short wires to both the center conductor and the shield to get them into the HD-15.
If you prefer, you can terminate each wire (you'll need 4 or 5, depending on if you use composite or separate sync) with BNCs and use a BNC/HD-15 breakout cable. This is the more professional way to do it, but it requires more expensive tools (to crimp the BNCs) and more "stuff", so it's more expensive all around. Make sure you use 75-ohm BNCs, not the (slightly more common outside of the A/V realm) 50-ohm BNCs. They appear virtually identical upon a casual inspection.
With proper cable, 0.7Vpp analog video at up to UXGA (1600x1200, VGA style signalling) can be run for several dozen meters without any issue. Just make sure both ends agree on the concept of "ground". You may need to run a dedicated (and heavy gauge) ground line between both ends. Consult an electrician before you do that as there are some issues associated with doing such a thing. If you're in the same residence (this is not always true of commercial buildings), there should be no problem.
Using proper transmission line that is matched to the source and termination impedance will eliminate transmission line artifacts that can show up on sharp edges. These artifacts would mostly look like "ghosting", similar to multipath interference on OTA NTSC signals.
For longer runs (>25m or so), consider using a balun and shielded CAT5 or CAT6 cable. Note that you need the balun for these longer runs on twisted pair cable! You will also have to use composite sync since there are only 4 pairs.