NOTE: This answer has been made at least partially wrong by updates. Exact numbers probably need fixing, and some methods might have been added that were not taken into consideration while writing this.
Get them to trade with one another! This may not be the best way, but it was the main tool for me thus far.
For trade to occur, both races need to be spacefaring and produce over 1000 of some resource - not the same for both races, of course. Then use either race's diplomacy menu to convince them to broker a trade agreement. For you, this is entirely free and requires no further involvement.
While the races exchange trade fleets (which they will be doing periodically until the trade agreement is broken), they will exchange 500 units of resources per month and receive a great bonus to their attitudes towards one another.
Not all races seem to be able to trade. I don't think Thoraxians and Burlusts can initiate a trade agreement (but will accept one), while Boarines seem to be altogether too antisocial to trade.
A more expensive, but quick method is asking Boarines to encourage relations between the two races. This requires a fairly high influence on the Boarines and costs credit, but the attitude boost is immediate.