What side did he take? Aaron Burr was never convicted of the crime so I am wondering how Jefferson handled it since Burr remained the official Vice President.
Let me start off by saying that Aaron Burr was not well-liked by his peers in Washington. Some historians call him the "Anti-founder" for his differences in personality and morality. I have a copy of my early American history textbook in front of me, Give Me Liberty! by Eric Foner, that sets the stage for Hamilton, Burr, and Jefferson's relationship:
"Before assuming office, Jefferson was forced to weather an unusual constitutional crisis. Each part arranged to have an elector throw away one of his two votes for president, so that its presidential candidate would come out a vote ahead of the vice presidential. But the designated Republican elector failed to do so. As a result, both Jefferson and his running mate, Aaron Burr, received seventy-three electoral votes. With no candidate having a majority, the election was thrown into the House of Representatives that had been elected in *1798, where the Federalists enjoyed a slight majority. For thirty-five ballots, neither man received a majority of the votes. Finally, Hamilton intervened. He disliked Jefferson but believed him enough of a statesman to recognize that the Federalist financial system could not be dismantled. Burr, he warned, was obsessed with power, "an embryo Caesar."
"Hamilton's support for Jefferson tipped the balance. To avoid a repetition of the crisis, Congress and the states soon adopted the Twelfth Amendment to the Constitution, requiring electors to cast separate votes for president and vice president. The election of 1800 also set in motion a chain of events that culminated four years later when Burr killed Hamilton in a duel. Burr appears to have subsequently engaged in a plot to form a new nation in the West from land detached from the United States and the Spanish empire. Acquitted of treason in 1807, he went into exile in Europe, eventually returning to New York, where he practiced law until his death in 1836." (pages 297,298)
So while I cannot speak for your specific question, how Jefferson handled Burr's killing of Hamilton, I can say that Jefferson did not like Burr personally. Like most early Presidents and Vice-Presidents, the two did not campaign together as one party, but were competing separately for the office of President, 2nd place receiving the Vice-Presidency. I hope this helps answer your question. I am interested if anyone brings forth any material that Jefferson wrote on the issue.
Jefferson was subpoenaed by Burr (there is some argument as to whether the subpoena went directly to Jefferson or by way of his federal prosecutor, but the effect is the same in either case) to release documents that would aid in his defense, as well as for him to appear in person. Jefferson refused to appear, and released only a few of the documents. Jefferson and Burr were already at odds politically, so he may have just been in no mood to help a political enemy, but Jefferson was not prone to petty politicking in life or death matters. Even when Burr was accused on very credible grounds of treason later on, Jefferson described Burr as "heretofore distinguished by the favor of his country" in an address to Congress, encouraging patience and the benefit of doubt when dealing with the matter. To my knowledge there are no known statements by Jefferson on his personal feelings about the duel, whether Burr was justified in shooting Hamilton -most duels of the period involved both parties intentionally missing, so that each could say they put their life on the line and leave with honor intact, and Hamilton expressed his intent to miss with his first and possibly his second bullet if it came to that-, whether Burr tried to miss and didn't, or any number of other opinions he may have had. Draw what conclusion you will from these facts, historians are divided on the reality of Jefferson's feelings on the event. It is pretty safe to say that a political ally and good friend, one to whom he almost certainly owed his presidency, being shot and killed by a political enemy wouldn't have sat well with him in any case.
Sources: Ron Chernow "Alexander Hamilton"
http://law2.umkc.edu/faculty/projects/ftrials/burr/burrjeffproclamation.html
http://founders.archives.gov/documents/Hamilton/01-26-02-0001-0241
Founding Brothers is a fairly accessible book that gives deeper insight of this event as well as events surrounding it. I would recommend it.