Constitutionally speaking, there is nothing which would prevent a presidential candidate from one party from selecting a vice-presidential running mate from the opposite party. A vice-presidential candidate must meet the age, residency and nationality requirements of a president, and cannot legally reside in the same state as the president. There is nothing which would exclude a Republican presidential candidate from choosing a Democratic running mate or vice-versa.
The political reality, however, is that both the Democratic and Republican parties prefer to run straight party tickets for the sake of unity and succession. A party's presidential candidate often seeks out a running mate who "balances out" the demographics of the country. This balancing act may include a conservative/liberal aspect, but to date it has not included a Democrat/Republican element. Each political party seeks control and influence over the Congress and the eventual nomination process for a new Supreme Court justice. A mixed presidential ticket may not send out a defined message of partisan control, since each party would have a very influential leader at the head of the executive branch.
Another reason a mixed presidential ticket might prove problematic is the line of succession. If a Democratic president should die in office, a number of party loyalists may feel disenfranchised if a Republican is allowed to assume the office without election. As moderate as a running mate of the opposite party may be, he or she would still be seen by many as a registered member of that party. Certain social programs or economic incentive packages endorsed by a former president may not survive under the leadership of the new one.
Originally, the president and vice president did not run together as part of a combined ticket. Before 1804, when the 12th Amendment was ratified, whichever presidential candidate got the second highest number of votes from the electoral college became vice president. John Adams, a Federalist, was vice president to Thomas Jefferson, a Democratic-Republican. Even after this, a few candidates have run with running mates from other parties; for his second term, Abraham Lincoln, a Republican, chose Democrat Andrew Johnson as his vice president. The practice of a mixed ticket is very uncommon, however.
In modern politics, a mixed presidential ticket might survive voter scrutiny if both candidates were seen as moderates in their respective parties. There has been talk in previous elections of a moderate Republican being approached by a moderate to liberal Democratic presidential nominee, but to date nothing has proceeded past the talking stage. While a mixed presidential ticket might be seen by the voting public as a sincere effort to bridge the gap between political parties, it might also be seen by party loyalists on both sides as a failure to produce a satisfactory same-party ticket or as little more than a noble but risky political experiment.
