All Pew Research Center surveys about U.S. politics and policy include questions about partisan affiliation. In 2014 a total of 25,010 respondents were asked the following:

In politics TODAY, do you consider yourself a Republican, Democrat, or independent?
(IF INDEPENDENT, OTHER, DON’T KNOW): As of today do you lean more to the Republican Party or more to the Democratic Party?

Detailed demographic breakdowns of party identification are presented here, or can be downloaded (CSV or PDF).
Report: A Deep Dive Into Party Affiliation

Among Non-Hispanic Asians