Can she ask her husband to use a condom to protect herself if he marries another wife?

Praise be to Allah.

Firstly: 

Zina is a major sin, great calamity and the worst of conduct, and its consequences in this world, in the Hereafter and in the grave are severe. 

Allah says (interpretation of the meaning):

“Made lawful to you this day are At‑Tayyibaat [all kinds of Halaal (lawful) foods, which Allaah has made lawful (meat of slaughtered eatable animals, milk products, fats, vegetables and fruits)]. The food (slaughtered cattle, eatable animals) of the people of the Scripture (Jews and Christians) is lawful to you and yours is lawful to them. (Lawful to you in marriage) are chaste women from the believers and chaste women from those who were given the Scripture (Jews and Christians) before your time when you have given their due Mahr (bridal-money given by the husband to his wife at the time of marriage), desiring chastity (i.e. taking them in legal wedlock) not committing illegal sexual intercourse, nor taking them as girlfriends. And whosoever disbelieves in Faith, [i.e. in the Oneness of Allaah and in all the other Articles of Faith i.e. His (Allaah’s) Angels, His Holy Books, His Messengers, the Day of Resurrection and Al‑Qadar (Divine Preordainments)], then fruitless is his work; and in the Hereafter he will be among the losers”

[al-Maa’idah 5:5]. 

Ibn Katheer (may Allah have mercy on him) said: 

Just as it is stipulated that women be chaste -- which means refraining from zina -- it is also stipulated for men; a man should also be chaste. Hence Allah says “not committing illegal sexual intercourse”, referring to adulterers and fornicators who do not refrain from committing any sin and do not control themselves. “nor taking them as girlfriends” means: limiting himself to one woman or mistress. End quote. 

Tafseer Ibn Katheer, 3/43 

Shaykh al-Islam [Ibn Taymiyah] (may Allah have mercy on him) said: 

If a woman's husband commits zina with another woman, and does not distinguish between halaal and haraam, his intimacy with that woman is the same as the intimacy of the zaani with a woman with whom he is committing zina, even if no one else is intimate with her, because one of the forms of zina is taking a girlfriend or mistress. 

End quote. Majmoo‘ al-Fataawa, 32/145 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: 

A woman saw her husband committing zina -- Allah forbid. What should she do? 

He replied: 

She should advise him, especially if that was the first time and she has children from him. But if he persists in doing that then she should seek an annulment of the marriage. However, in general, she should weigh up the pros and cons and decide on that basis. End quote. 

Thamaraat al-Tadween min Masaa’il Ibn ‘Uthaymeen, p. 112 

See also the answer to question no. 115107 

Secondly: 

The wife does not have the right to ask her husband to use condoms during marital relations, unless there is a reason for doing that. The fact that he has married another woman in a legitimate shar‘i marriage does not make it permissible for her to do that, unless it is clear that the husband himself has a disease, such as AIDS and the like, which could be transmitted to her through sex, or it becomes clear that the other wife has a disease of this type, or the husband was in a haraam relationship outside of marriage. In that case she has the right to ask him to use condoms, so as to ward off the harm that is thought to exist in him, until it becomes clear that he is free of that. If it becomes clear that he is healthy and there is no obvious source of danger of transmitting disease through him, then she no longer has the right to ask him to do that. If it becomes clear that he does have a disease which could harm her or be transmitted to her, then she has the right to ask him to carry on using it; indeed, in that case she has the right to ask him to annul the marriage, if his sickness poses a danger to her and it is something that it is difficult to treat or avoid, such as AIDS and the like. 

See: al-Ahkaam al-Shara‘iyyah al-Muta‘alliqah bi Marda al-AIDS, by Dr. ‘Umar Sulaymaan al-Ashqar, in Diraasaat Fiqhiyyah fi Qadaaya Tibbiyyah (1/25 ff). 

And Allah knows best.