Furnace is Sweating an Abnormal Amount
There are two main contributing factors at play here. First and most importantly, the evaporator is getting too cold. The other problem is only a problem, because the first problem is a problem.  And that is, that the humidity in the basement may be too high.
What's happening, is that the box that contains the evaporator is getting too cold. Once it cools below the dew point, it causes the moisture in the air around it to condense. To solve the issue, you're going to have to figure out why the evaporator is getting so cold.
Improper refrigerant level
My first guess, would be that the system is not charged to the proper level. Specifically, in this case I think it may be undercharged.  Though, you may also see this if it was overcharged. However, since it sounds like it hasn't always been a problem, I'd suspect the system developed a leak.
Testing
To test this, you're going to need a set of gauges. If you don't have a set; or don't know how to use them, you're going to want to contact a local HVAC company.
Restricted air flow
If not enough air is moving through the coils, the little bit of air that is there could be getting too cold.
Testing
Check the filter, to make sure it's clean. Check the evaporator coils, to make sure they are clean. These are the two most common causes of restricted air flow. Additionally, make sure registers and returns are open and unobstructed. If your system has a multi-speed blower motor, make sure it's set to run at the manufacturer's recommended speed.
Long running system
If it's really hot outside, and your system is running almost constantly. It could just be that the evaporator doesn't have time between cycles to warm up.  If this is only a problem when it's abnormally hot outside, it could be &quot;normal&quot; for that situation. Typical A/C systems are designed to cool between 20-30°F, anything beyond that and they'll struggle to keep up.
High humidity
If everything is operating normally, but the system just happens to be in an area where the humidity is high. It might be a good idea to insulate the system, especially in the area around the evaporator.  This is common when the equipment is installed in unconditioned spaces (attics, garage, crawlspace, etc.), but if your basement has moisture problems, it might be required there too.
If the moist air cannot get to the cold metal, the moisture in the air will not condense. However, if the now insulated system gets too cold, you could have problems with the evaporator freezing. So I'd only install insulation, if I was sure the system was operating normally.