The gemarah on 45 B , says that you are allowed to use the bath house or garden of a Avodah Zarah as long as " Shlo Betovah ", you don't give them any benefit. But you are not allowed to use it, if you give them some type of benefit. From this , I imagine you can infer that you are allowed to use the basketball court, softball field of a church, or even use their yard as a short cut to the other side of the street, as long as you don't pay them for the use, or any other type of appreciation ( and of course, when there are no issues of Maris Eyen, i.e.., on a day when the church is closed ).
The question arises, is that it is fact, that religious people use the pool at the YMCA ( a Catholic organization ), and pay to use it. Clearly benefiting the Catholic religion, as opposed to using the pool at Jewish organizations. Now if you might answer that the Christian religion is not Avodah Zarah ( as can be interpreted in the Rema O.C. 156 ), that only applies to Non-Jews. To Jews it definitely is Avoda Zarah, and would be forbidden to support such organizations. Can you please explain to me the Heter or Melamed Zecut, on those who permit this practice. And if so, can this also be applied to other things, i.e.. can we pay the church for use of their facilities ? Thank You.
Rabeinu Tam learns that when you pay it is permitted, and it is only prohibited when you have to actually be Makir Tov to the Avodah Zarah. The Rema (YD 143:3) says that although we are Machmir like Rashi when the Avodah Zarah or the priests make the profit, we can be lenient if the bathhouse is not in the churchyard itself.
However, all of this applies when the business belongs to the church itself or to the priests. However, I do not think the "Y" actually belongs to the church or to the clergy; rather, it is merely affiliated with the religion. If this indeed is the case, then since the profits do not go directly to the church (even if they do benefit indirectly), it is permissible.
While my conclusion is in agreeance with you, Halachakly. Rav Moshe, zt'l, writes in O.C. IV siman 40 ,paragraph 26 ( page 69 ), that this is not a good idea due to missionaries that are present in such places. And there is definetiley a missionary presence in such places.
If indeed there is a missionary presence there, then it certainly should be avoided. Rav Moshe Shternbuch (Teshuvos v'Hanhagos) says the same with regard to the use of a missionary hospital for giving birth, and says that the missionary presence creates a problem. However, he writes that for a frum person with a solid grounding in Emunah, it is not Asur m'Ikar ha'Din to take advantage of the service of the hospital (since it is much cheaper than other hospitals), but one should not do so because other Jews, who do not have a strong Emunah, will also go there and might be led astray.
I do not believe that you can answer the question if you use the terms catholicism and christianity interchangeably. R' HR' Moshe Feinstein, z'l addressed this very clearly in a tshiva regarding attendance at alcoholic's anonymous meetings. Particularly that if it was the only possibility that an x-tian church is allowable, eastern orthodox x-tian church questionable, roman catholic church un acceptable. Obviously this is based on the presence and perception of statues, as well as other theological beliefs that catholics hold different from x-tians.
How does this apply to Non-Frum organizations that have similar types of centers. People pay money to these places and this funds not avodozara but I would think Anti-Pro-Torah views.
As mentioned above, if it is known for certain that one's money is being used for purposes of Hasasah (causing Jews to stop doing Mitzvos), then certainly one must not give his money to such an organization. Obviously, one must use discretion, and should bring any question to a competent rabbinic authority.
Below are relevant excerpts from a letter written to the Editor of HaModia newspaper on the topic of using the facilities of a YMCA, sent to us by the writer of the letter in response to Shmuel K.'s question above:
I called... Rav Fisher Shlita and posed the question myself. While he said that according to the data that was presented to him then (almost 15 years ago - that they only rent the space from the YMCA, which I've since found out is not true, but that the studio is in fact a full program of the YMCA), he maintained that one is not technically transgressing a law of supporting a place of avoda zara. HOWEVER (even it was only a rental situation, which it is not), lifnim meshuras hadin, one should refrain from attending such a place. AND MOREOVER, if there are other alternatives offered by establishments owned and operated by frum Yidden, then certainly there is no question that one should not patronize such a place.
... The pasuk says: "When you buy or sell goods or services from your colleague (ameesecha)". The Minchas Yitzchak brings down (Chelek 3, Siman 129), quoting from the Chasam Sofer and others, that we learn from this pasuk that wherever possible, one should patronize the businesses of fellow religious Jews, as Ameesecha is understood to be "Am she'eetcha beTorah UMitzvos", the people who like you follow Torah and Mitzvos. This is also stated LeHalacha in Teshuvas HaRama, (Siman 9), where he implies that this holds true even when he can find something cheaper by the non-religious or non-Jewish establishment.