For various historical, sociological, and spiritual reasons, Jewish law has, for more than two millenia, determined that Jewishness is determined by the birth mother. That means if a Jewish woman and a non-Jewish man have a child together, the child would be considered Jewish by the standards of Jewish law. Consequently, it also means that, if the situation were reversed, and a Jewish man has a child with a non-Jewish woman, the child would not be Jewish. Therefore, in your situation, any children you have with your Jewish partner would be Jews by birth, even if you had not yet converted.
All demographic indicators point to the reality that, from the perspective of Jewish continuity and Jewish commitment, Jewish children fare better when both parents are Jewish. However, there are plenty of good examples of Jewish children who grow up with one Jewish parent and go on to live dedicated, engaged, meaningful Jewish lives in adulthood, passing on their deep Jewish connections to their children as well. By the way, the opposite is true, too: there are plenty of Jewish children who grow up with two Jewish parents and don't have strong Jewish ties in adulthood. The result in all cases is largely dependant on the approach of the parents. When parents, even in households with one Jewish parent where the non-Jewish parent is supportive, show their children that being Jewish and living Jewishly matters to them, when they demonstrate Jewish ways of living and practicing at home, when they prioritize Jewish education, when they embody Jewish values, the children are more likely to stay Jewishly engaged. When parents don't, even in a household with two Jewish parents, they are unlikely to be successful raising committed Jewish kids.
In a case like this questioner's, where he and his partner are striving to be the kinds of parents where Judaism is fully present in their home and in their family life, and since their children would be Jewish whether or not he converts first, I see no Jewish problem with him trying to start his family before he converts. Indeed, the first command in the Torah is "be fruitful and multiply." We are only granted a limited window in our lives to fulfill this mitzvah. Becoming Jewish in a manner that reflects seriousness and commitment takes time, and is not biologically-restricting in the same way.
On the other hand, it is important to bear in mind that converting to Judaism can be a major time commitment. So is raising children while juggling all of life's other necessities. If one has kids before he converts, he may find that he simply doesn't have the time to pursue conversion (cf. Mishnah Avot 2:4, "Say not, 'When I have free time I will study. Perhaps you will never have free time!"). Therefore, if it is possible to convert to Judaism before one has children (in a timeframe that would not inhibit one's ability to have children), I would strongly advise that approach.
Answered by: Rabbi Michael Knopf