I'm amazed that any of these are still out there. They were great monitors but the DA board is a ticking time bomb and I figured most of them would have been scrapped when they stopped working.
Sony never made the GDM-20D11 service manual available to the public. The GDM-20SHT is the closest model with readily available service literature.
The first thing to do is replace all the surface mount capacitors on the DA board. Twist them off with pliers then desolder what's left of the leads.
If your DA board is A-1341-684-D then you need 27pcs capacitors:
1uF/50V ⌀4mm 9pcs
2.2uF/50V ⌀4mm 2pcs
4.7uF/35V ⌀4mm 1pcs
10uF/16V ⌀4mm 1pcs
10uF/35V ⌀5mm 2pcs
22uF/16V ⌀5mm 6pcs
47uF/16V ⌀6.3mm 6pcs
The earlier A-1341-684-C board may have one fewer 2.2uF/16V capacitor.
While you're in there, you should ESR test all the capacitors on the D board and replace the bad ones.
If after this you get good video but it's still dim and slow to show a picture, I would check if the heater voltage is less than 6.3VDC (be careful, there is high voltage on other CRT neck pins) and if so, check C661 and C664 on the G board next to the 4-terminal TO-220 regulator IC655, and check the IC655 soldering and traces for hairline cracks. I don't think I've seen this problem on this monitor, but it happens on the other 20" models.