Under Article I, Section 5 of the U.S. Constitution, it would seem that a house of Congress has the power to determine whether someone elected to a seat is qualified to be seated - it is "the judge of the elections, returns, and qualifications of its own members."
Still, the U.S. Supreme Court held in 1969 that a house of Congress cannot "exclude" an incoming member meeting the Constitutional minimums, but must "expel" him. But...there's also earlier precedent out there in practice for a house of Congress refusing to seat a duly elected Congressman.
As in the case of the first Socialist elected to Congress, a man barred from being seated not once, but twice in the wake of World War I - the mustachioed Milwaukeean, Victor Berger:
In 1911 Milwaukee sent him to Congress as the first Socialist representative.* Appalled, Washington gaped at this ''political monster" who was no monster at all but a round German burgher, bald, shuffling, infinitely good-natured. Re-elected in 1918, he was refused his House seat by a vote of 309 to 1 because of his pacifist doctrines. In 1919 he was again elected, again barred.
A more recent SCOTUS case, Nixon v. United States, while distinguishing and officially retaining Powell - the 1969 case, arguably indicates a greater willingness of the Court to defer to a house of Congress conducting its own Constitutionally enumerated internal affairs. However, the 1993 case deals with impeachment, not exclusion of a member, so Powell's bar on barring would seem to stand.
Interestingly, Senate Majority Leader Harry Reid's letter to Blago on behalf of the Democratic Senators invokes a power that the Senate arguably does not have under the case law in Powell:
Please understand that should you decide to ignore the request of the Senate Democratic Caucus and make an appointment we would be forced to exercise our Constitutional authority under Article I, Section 5, to determine whether such a person should be seated.
538 further speculates that because a Blago appointment would be just that - an appointment - it would not be included in the Senate's A1,S5 power to judge "elections" of its own members...meaning, in theory, the body would have to permit the new member to be seated and then get a 2/3 vote to expel the new member. I think the "qualifications" bit would be the Senate's stronger scrap of text, though. The Senate hasn't expelled a member since 1862 (for supporting the Confederacy).
*I'm going to chalk this post up to tangential studying for Foreign Affairs and The Constitution.