The humid weather is resulting in "sweaty" cold water pipes in our
basement.

I have read the suggestion to insulate the pipes just like with hot
water pipes. However, I am concerned that while this may reduce the
sweating, there still will be some condensation, but now the
condensation will be trapped against the pipe.

Should I be worrying about this? If so, what should I do to minimize
the potential for mildew and mold in this trapped moisture?

Second, I find that the condensation only occurs on the first few feet
of pipe in from the grounde. Is it possible that conduction causes the
water in the first few feet of pipe to stay cold all the time leading
to heavy condensation while beyond that the water for the most part
has warmed up (except when the water is actively running) so that
there is much less active condensation?