This Adida et al. (2019) paper used the same design as Hainmueller and Hopkins (conjoint experiment) for Syrian refugees and their study was in the field during Oct. - Nov. 2016. Compare that to the 2011-2012 sample from Hainmueller and Hopkins (publishing results can take a while). I was unable to find a conjoint experiment for immigration attitudes more recent than fall 2016. Of course, it is possible that if Trump affected what Republicans and Democrats value in potential immigrants (in this case, asylum-seekers) then we would see the effects by then after more than a year of campaigning for president.
As one would expect, they find that Republicans are much less interested in granting asylum than Democrats. However, it seems that Democrats and Republicans similarly value the ability to speak English, certain professions, sex, and religion. For instance, both Democrats and Republicans prefer Christian asylum-seekers to Muslim ones. Generally, what Democrats prefer Republicans also prefer, even if the absolute levels of support are very different across party (Fig. 9). Said another way, their results suggest that Democrats and Republicans disagree about how many immigrants (asylum-seekers) should be admitted --- but if they had to choose they would probably rank the potential immigrants similarly.