Dowries were an expected part of marriage and marriage negotiations in Eastern Europe, so quite common.

Dowries, the property brought by a man and a woman into a marriage, are referenced in the Hebrew Bible, and laws and customs regarding dowries and marital property were expanded upon in the Talmud and then codified in Jewish law, so it is quite a longstanding practice in Judaism. The ins and outs of all of the laws regarding dowries is very long and involved, but on a very, very basic level the bride was expected to bring some assets into the marriage, which she could recover on the occasion of divorce or her husband's death.

In Eastern Europe, traditional prenuptual agreements between the couple's parents consisted of the dowry (Yiddish: nadn), assets that the bride would bring in, and what was referred to in Yiddish as kest, when the family of the bride (or groom) would pledge to support the couple for a proscribed period of time by providing them with a place to live, and/or financial support so that the groom, if he were so inclined, could study in yeshiva. Once the sets of parents agreed on terms, a document laying them out (tenoyim) would be signed at a betrothal ceremony (these days traditional Jewish wedding ceremonies include both the betrothal and the marriage; it used to be that these were two separate ceremonies that could take place months apart). Obviously the more the bride's family could offer in terms of the nadn and kest, the better they could attract a good match (roughly speaking, defined by the groom's lineage, Torah knowledge and study, and family wealth). Of course, not all families could afford this, and communities usually had a bridal fund (hakhnoses-kale) to provide poor or orphaned brides with a dowry (fundraising for bridal funds exists in ultra-Orthodox communities to this day).

As a traditional Jew living in the shtetl, Tevye would have been expected to provide a dowry for each of his daughters. Given his relative poverty, and the number of daughters he would have to marry off, it makes sense that the dowry, and the types of men it would attract, would be on the daughters' minds. In fact, though the number of daughters Tevye has in the original book, Tevye the Dairyman (Tevye der Milkhiker) is inconsistent (5 or 7), the background for Tevye having 7 daughters is the Yiddish saying "seven daughters is no laughing matter" (zibn tekhter is nisht kayn gelekhter)-precisely because of how difficult it would be to provide them all with sufficient dowries to make good matches for each.

I know that this is not the point of your question, but I really want to put in a plug for reading Tevye the Dairyman by Sholem Aleichem, and/or at least the Yiddish film (with English subtitles) Tevye. I get why people are into Fiddler on the Roof, but the book it is based on-and the Yiddish film from 1939-are so much richer and beautiful (and heartbreaking).

There are a few sources for this, including various works from the Jewish Enlightenment objecting to this practice and Jewish legal texts, but for some basic background: Yivoencyclopedia.org/article.aspx/marriage

Ruth Wisse wrote about Tevye (I can find the exact source if you'd like, though if you google her name and "Tevye," you'll probably find it)