Indoor relative humidity over 50% can cause problems. High humidity from A/C operation typically arises from having a one of two causes:
[1] The unit capacity is substantially oversize for the space being cooled.
In this condition, moisture is removed much more slowly than the temperature set point is achieved, such that the unit runs for short periods, cycling off before removing much moisture. To check this, compare the square footage being cooled to the unit cooling capacity. Capacity usually should be 12000 BTU for every 400-600 square feet (400 for old poorly insulated homes, 600 for higher efficiency homes).
[2] The unit has an evaporator temperature that is a little too warm.
To get good moisture removal, the evaporator temperature must be less than 55F. Upper 40s to 50 works well. Using a thermometer, check the discharge temperature at a register near the unit when the A/C has be running continuously for 5-10 minutes (turn the thermostat way down). Warmer than 55F probably means your evaporator temp is too warm to dehumidify effectively. Evaporator temp too warm usually indicates either too much blower for the compressor capacity (select a lower speed) or the system is a little undercharged (to correct this on an older R22 system you need a licensed tech). There are other more serious underlying causes like the compressor wearing out. If the system is generally OK, you can safely select a slower blower speed to lower the discharge air temp. If you do this make certain the discharge air leaving the coils is not colder than 25F below the inlet and that no frost forms at the refrigerant liquid inlet side of the evaporator coil. (This is an approximation. The accurate professional way to do this is with a set of gauges.)
For the curious:
Why no colder than a 25F temperature drop at the evaporator? If the house is 72F, then the discharge air temp at the evaporator would be 47F (25F colder). Compressor mfgs require 15F superheat in the vapor returning to the compressor. Roughly speaking, 47F air temp => 50F vapor temp of refrigerant returning to compressor (the evaporator is a cross-current heat exchanger so yes, the vapor temp to the compressor can be warmer than the air leaving the coil) => 50F vapor less 15F superheat means there is a 35F saturated suction temp. The saturated suction temp must be a few degrees above 32F or the unit will collect frost and turn into an ice ball instead of taking liquid moisture out of the air. By the way, if you lower the blower speed and the underlying cause of the problem is that the unit is low on refrigerant, then the unit will be less efficient than with the correct refrigerant level (it will use more electricity to get the same job done). On older units, the condenser temperature split should be 25F-30F. When the outdoor temp is at least 80F, measure the air temp going into the condenser and measure the air leaving the condenser. The air leaving should be at least 25F warmer. If not, then your system is probably low on charge or the compressor is worn.