I won't comment on whether Eisenhower was a good or bad president in general terms, but more specific to your question regarding the President and the GOP.

A number of factors contributed to the drop in GOP numbers over the Eisenhower period. I'd suggest the biggest being the three (brief, and relatively mild) recessions during his time in office. Another factor was Eisenhower's problematic relationship with civil rights and domestic spending on "welfare" programs: simplifying, he was against them. The reason this was a problem was that while the Dems held strong in the South, Eisenhower's being basically a racist penny-pincher slowly ate away at the conservative coalition and began breaking the GOP's strength in the Northeast and the West coast to more progressive minded Dems. For example in 1958 the GOP lost 48 in the House and 12 or 13 (I can't remember the exact number, apologies) in the Senate.

Another significant factor in this was that Eisenhower was not really a "good" Republican - his party allegiance was basically a last minute revelation, he loathed campaigning and rarely did much at all to support the party during midterm elections (sending out Nixon instead and then blaming Nixon for the seat losses), and was indignant whenever Republican politicians acted in self-interest rather than national interest (as defined by him).

Basically, whatever your perspective on whether Eisenhower was "good" for the U.S., he was not really a good thing for the GOP, and he really did not try to be. This is a criticism that comes up in a good deal of the literature on Eisenhower (Leuchtenburg, Newton, et al.), that he typically refused to use his immense personal popularity for the benefit of programs or policies that might have helped the nation, the GOP, or the Cold War.

Hopefully this is helpful. Keep in mind others might well disagree with my interpretation.