Where does the basement drain go? A municipal system? Septic tank? Cesspool? (If any are still in use.) And are you sure it's not coming up around the drain, but not from the drain?

My strongest suspecion is that drains are improperly trapped and/or very short before opened to daylight. In addition to traps on lines,some arrangements must not have an open tee on lines. If you see an inverted T in the lines,temporarly plug the opening to see if that cures or improve's leak. The inclosure must be reasonably air tite which can be helped with foil tape. As air flows through handler,which does it pass through first,secound and third ,blower ,filter, evap coil? Was blower,coil and strip heaters replaced as a unit or was part of old handler retained? If this turns out to be the problem a baffle can be installed on condensate pan as last resort. By rook or crook,you must have a filter or things will just get worse.

you may use test plug on the floor drain or cut it off and cap it. Then cement over it. For the utility sink drain cut it off below the floor and cap it, Fernco has rubber caps for varying pipe sizes. These should be readily available at any home improvement store

This may not be a clog problem at all. The house is 55 yrs old which tells me that the line in question may be an 1-1/2" cast iron drain line. In the mid to late 80's the appliance industry began using a higher volume pump in washing machines, which in turn prompted the plumbing industry to change the standard size of a laundry standpipe from 1-1/2" to 2". To the untrained eye this may appear as a minor change but in fact a 2" line has nearly twice the volume of an 1-1/2" line. 1-1/2" cross sectional area = 1.76sq.in 2" cross sectional area = 3.142cu.in. In the final analysis I think what you will find is that the size of the line itself may be marginal for the discharge rate of modern laundry appliances, which in itself would result in a minor backup as the machine pumps out. This problem is further complicated because the original pipe is most likely cast iron. While cast iron is a fantastic material for DWV piping it has one minor drawback. In its new state the interior walls of cast iron pipe are comparatively rough, which means a high coefficient of friction. This is further complicated by the fact that the pipe will rust slightly as it ages which in turn further increase the friction loss. This was further complicated by habitually cleaning tools and buckets and discharging the residue into the drain, which no doubt has by now caused a severe buildup of foreign matter in the pipe which cannot be removed even with a powered auger. At a minimum the line will need to be cleaned with a good heavy duty powered auger with the proper sized cutter head, and even that may prove to provide marginal relief. We are then left with two options as a permanent solution. 1. The preferred solution would be to replace the existing line with 2" PVC. An even better option would be to opt for the exception in the Plumbing code which allows us to increase a pipe diameter by one nominal trade size. Understanding that the code now requires a 2" line for a laundry standpipe, the option would then allow us to increase the line to 3". The second option is much simpler although sometimes less desirable. You could discontinue the use of the laundry standpipe and install a laundry sink beside your washing machine. You then discharge the washing machine into the sink. A laundry sink is only required to have a 1-1/2" trap and drain line. In this configuration the sink serves as a reservoir for the discharge from the washing machine, then the 1-1/2" drain line from the sink will meter the flow into the existing 1-1/2" drain line at the proper rate to insure good flow under gravity flow conditions without causing the lines to back up.