Yes, they did. Rabbinic literature was not exactly on the bookshelf of every Christian theologian, but it was not uncommon for ideas to pass that way.

Perhaps the most famous is Maimonides. Besides his legal code (which was probably his most important work in Jewish circles), he also wrote philosophy, most famously in Guide to the Perplexed. It discussed religious issues in tandem with Greek philosophy, intended mostly as a "Jewish response to and version of Hellenistic philosophy", as opposed to the more common outright rejection of it. It eventually circulated around to Christian communities. While Jewish theology does still end up wildly different, and certainly no Christian theologian could read Maimonides with complete agreement for long, occasionally there were topics on which Christian theologians would take influence from Maimonides' ideas.

Another area was biblical interpretation. Writing commentaries on the Torah was an important part of Medieval Jewish scholarship--there are loads of them. One particularly important one in this context is that of David Kimhi (usually known in Jewish contexts as the Radak, which is simply an acronym of "Rabbi David Kimhi". Most Medieval Rabbis have similar naming conventions--Maimonides' is the Rambam). His Torah interpretation was translated and circulated outside the Jewish community to a greater extent than others'. Because these commentaries usually synthesize previous Jewish religious scholarship on an issue, while adding the commentator's unique philosophical bent, reading a commentary like that can give a reader access to a significant part of the Jewish understanding of a religious topic, beyond a particular Rabbi's opinion of it.

When, and at what level? The reason I ask is that it is fairly common now for some ideas from at least the Mishnah to influence Christian theological thinking, but more at the level of single issues rather than grand ideas like the Trinity or predestination.

To give a topical example: the traditional Christian interpretation of the fall of Sodom was that they were destroyed for sexual sins. While this is not stated explicitly in the Hebrew Bible / Old Testament, the Qu'ran does state it explicitly (so I gather - I haven't read the Qu'ran to check). However the Mishnah interprets the incident as being about the sin of inhospitality (the citizen of Sodom attempt to attack the angelic visitors to the city). This contrasts with the hospitality of Abraham (who offers the visitors food earlier) and Lot (who protects the visitors, and is prepared to do so at the cost of his daughters). The Jewish interpretation has become more popular among Christian theologians and preachers in recent years. However you can see that while this may be important in current debates over sexuality, it doesn't affect any of the traditional major questions of theology dealing with the nature of God, and His relationship with humans.