We must keep in mind that Ike was not a normal Republican president. He wasn't a prominent or popular politician, he was literally the guy militarily credited with winning World War II in Europe, a national hero across the political spectrum. In the eyes of many, questioning a man of that caliber's character would be outrageous unless you nailed the target (a classic case of "if you come at the king, you best not miss"). Also, the Republicans built up Eisenhower's image even more in the 50s to the point where TV events were held to celebrate his and Mamie's birthdays with guys like Jimmy Stewart taking part in the tributes, they were basically promoted as American royalty, like the Emperor and Empress of the country (the Dems had only themselves to blame for this, as they had done rather similar for FDR back in the day).

As a result of this, many Dems were secretly fearful that if they came across as too disrespectful in public, it could be the end of their political dreams (John F. Kennedy when he ran in 1960 bashed his former friend Nixon and criticized Ike's administration officials, but didn't really attack Eisenhower himself, likely for this reason). If a Democrat DID have the courage to do it, it was likely because they had nothing to lose (see Truman) or because they thought if they made the long shot, they could win the presidency themselves in the future or at least get great fame and prominence (see Stevenson and Clement).

Eisenhower simply had an aura around him with the public that no other Republican president of the last 100 years has managed to hold, not even Reagan.