I don't see any conflict of interest ... and in fact, if DD gets the position (which I presume is a competitive application process itself), and impressees the Senator , it would be a nice ticket when applying for nomination from that MOC. I would fully expect that Senator to nominate (or at least give a competitive nomination to) one of his/her interns.
The potential downside is the impact the internship has on other nomination prospects, ie.. if your state has a split delegation with representatives from both parties , a MOC from the other party may view the internship as a negative. That's unfortunate, as the internship is a good opportunity regardless of which party DD interns for, but in today' s political environment ,there is some risk.