Warm humid air is finding it's way to the cool metal where it condenses. Remove cover,fill void with insulation and/or foam,replace cover and you are rid of it.

I'll agree that you've got condensation on the metal grill. I'll also agree that insulating any voids around the duct would likely remedy the problem-- if it were occurring in an above-grade living area with a path for unconditioned air to come in contact with the cool grill such as ductwork in an attic directly above. A basement is a different scenario.

Basements tend to have much lower heating and cooling requirements than above-ground living areas. If the HVAC system isn't a zoned system, a thermostat in a warmer area (e.g., the 1st floor) can be calling for air conditioning, introducing cool air into the basement that wouldn't otherwise be calling for it. My guess is that the basement stays around 65F +/- without conditioning during the month of May. So, introducing air that would be around 50F when leaving the evaporator coil (and close to saturation) increases the dewpoint of the basement air (and the relative humidity.) Stated otherwise, your air conditioner is pumping moisture into the basement. Let's do the numbers....

Weather data for Washington D.C. for May 2024 from Weather Underground https://www.wunderground.com/history/monthly/us/dc/washington/KDCA/date/2024-5 indicate that dewpoints ranged from 58.6F to 70F month to date. That entire range of dewpoint temperatures is higher than the surface temperature of the metal grill.

There are a couple of alternatives to consider including installing a dedicated system for the basement, zoning the existing central system, installing a duct heater or installing a dehumidifier. If you go the dehumidifier route, I recommend integrating one into the HVAC system versus a free-standing unit. Find a competent local HVAC contractor to take you through the pros and cons of each so you can make an informed choice.

Lack of insulation and air sealing is always the culprit with condensation.

Absolutes can absolutely get you in trouble. While poor air sealing and/or insulation can, and often does result in condensation, consider the potential for humid air to enter even a code-compliant (<5 ACH50) air sealed home due to stack effect in the summer. Then consider what happens when ambient dewpoints are greater than the LAT from the HVAC system's evaporator coil. Add to that humidification of the basement area which results from moisture migration through basement walls and you've got a great recipe for condensation.

The D.C. area experienced significantly higher than normal rainfall during the months of January, March, and April 2024. Stay tuned for May, but that looks likely, too. That mIght result in more moisture infiltration in a basement. Plus, we've had some really high ambient dewpoint temperatures. Maybe the perfect storm....

The OP can check the relative humidity and temperature in the basement using an inexpensive thermo-hygrometer that can be purchased in a big box store for about $10. With than information, the dewpoint temperature can be easily inferred.

Condensation, by definition, involves moisture condensing on a surface at or below the dewpoint temperature (of the surrounding air.) The temperature of the supply register should be nearly that of the evaporator leaving air temperature, which would be 50F +/-. In our coastal Virginia climate, we often see condensation form when warm, humid attic air comes in contact with supply registers connected to ductwork located in attics. We also observe it when a door is left open for a prolonged period of time during the hot, humid summer months--especially if there is a two-story foyer. Warm, humid air, which is less dense than dry air, rises to the upper levels in a home and condensation occurs on any surface below the dewpoint--usually supply grills.

The fact that the entire face of the supply register is wet suggests that it is not moisture draining from the interior of the ductwork.