Running the fan continuously CAN increase the summertime indoor humidity if the ducts aren't well balanced (with a correctly sized return for every doored-off room with a supply register) &/or the the house is air leaky. For duct heat/coolth delivery to happen pressures are induced in the ducts, and if a room is pressurized or depressurized relative to the outdoors it drives outdoor air infiltration. With a properly balanced duct system the pressure differences are small, and the air handler driven infiltration levels are small.
An Energy Star duct system would measure <3 pascals (0.012" water column)pressure differences between rooms under all operating conditions, doors open or closed. Most pre-existing duct systems will fail this test.
Only if the system is way oversized (sadly an all to common issue) would the re-evaporation of the moisture off the coils come in to play. This is really a "failure to dehumidify" problem rather than adding-moisture problem. In general right-sizing the equipment (or better still, right sizing MODULATING equipment that runs almost continuously) does a better job of both cooling the upper floors and keeping humidity levels bounded.
Running the blower continuously also chews through electricity adding HEAT to the house rather than cooling it. With properly designed ducts and running at low speed an ECM blower can run continuously without a significant heating effect.
If yours is a not-so-smart oversized 1 or 2 stage system you may get better bang/buck out of one of the U-shaped modulating Midea window-shakers for auxiliary cooling upstairs.
The 1-tonner only uses ~100 watts when running at it's minimum compressor & fan speeds in normal cooling mode (that is a fraction of the blower power on a central air handler), but pulls about 1100 watts when ramped up to full tilt (which won't happen if your central air is designed reasonably and running.) For those days when the sensible cooling loads are low it can be run in "DRY" mode, where it draws 300-500 watts (about the same as a high efficiency blower on a whole house system), but rather than heating the places it's removing humidity and doing at least some sensible cooling with that power. This is how I'm currently dealing with humidity & comfort issues the ludicrously oversized 1-speed low efficiency central air (a ~10 SEER 5 tonner for a <2 ton 1% design load) that never runs long enough to really cool the upstairs. At 15 CEER this quiet modulating window shaker is more efficient than a code-min efficiency central air unit too, and with no ducts to balance, it isn't driving infiltration.
They aren't cheap. In short supply earlier in the summer and there was and still is a lot of price gouging going on. I was looking for the 8000 BTU/hr unit, but found a 12,000BTU/hr unit online for less money than what the Walton family was (and still is) gouging for the 8000 at their box stores. (MSRP is ~$350 on the 8K, ~$450 on the 12K.)