A comment has prompted me to provide an answer to this question.
1. Of their nature, trade unions contain a range of different views. This is because they are "political" organisations (with a small "p"). After the various political activities involved, a trade union might well adopt a particular policy position. However, that does not mean that everyone in the union supports the policy, and the politics involved bear witness to that. So just because I am a member of a certain union, it does not follow that I support every single one of its policy positions, and people do not/should not assume that.
2. Politics has aspects of collaboration with others of different views, and of trying to achieve what might be possible in imperfect circumstances. Many decisions about particular courses of action are primarily political - in other words, about what is the best way of a gaining a particular outcome. The outcome being sought might well derive from other principles, but one might well keep a low profile on one issue while you are working behind the scenes on another. Staying involved gives the opportunity to influence.
3. There do come points where one needs to make a stand. I, for example, resigned from the Executive Committee of my union when they adopted a policy paper on equality for LGBT staff and pupils in all educational settings, including pre-school settings. I did a news release that got some limited coverage in the Catholic press at the time. In various discussions with the then President and a newly appointed General Secretary, no-one could be under any illusions about my position. Three or four years later, at our recent annual conference, people still very clearly remembered the reason for my resignation. However, one can make a stand through writing letters to the union magazine, the General Secretary etc. This is what I now do.
4. About half of my work as a Branch Secretary is supporting or advising members with regard to their working lives. I see this as very much a part of my lay apostolate in the Church - using my professional expertise to work for justice for colleagues. My experience of casework is such that I would not now dream of being in a work place without access to the support of union membership.
5. Another, more general, view of things is to see union membership (and engagement with the union's processes) as part of dialogue between Christian faith and the world. Dialogue does need engagement, and membership allows that. At our recent conference I had an encouraging number of conversations with other Christian delegates.
On a more personal note, my mother was a trade union branch official in her youth. This was prompted by her membership of Young Christian Workers. So it never occurred to me to do anything else except join a union when I started work. What I do find amazing at school is just how unaware pupils are of trade unions and the work that they do.