Answer

See Responsa Yechaveh Da’at, part 1, siman 38:

Question: Is it permitted to swim in the sea or in a pool during the week in which Tisha B’Av falls?

Answer: In tractate Ta’anit (30a), it is explained that according to the basic Jewish law there is no prohibition on bathing even on the eve of Tisha B’Av, until the time of the final meal before the fast. However, the Jewish custom is to be stringent regarding bathing even before Tisha B’Av, as explained in Sefer HaRokeach (siman 312), that Rabbenu Kalonymus of Rome ruled that it is forbidden to bathe beginning from Rosh Chodesh Av. So too it is written in Sefer Pardes HaGadol (siman 156), and in Sefer Shibolei HaLeket (siman 264). And it is clear that the intent is not to forbid bathing from Rosh Chodesh Av as a matter of law, but rather as a matter of custom, as they said in Nedarim (15a): regarding things that are permitted, but others adopted the custom to treat as forbidden, you are not permitted to allow them in their presence.

As for swimming in a pool or at the seashore, where there is no concern at all of immodesty or of mixed men and women, from the words of Maimonides and Nachmanides, who prohibited only entering a bathhouse during the week in which Tisha B’Av falls—that is, bathing in hot water—it appears that cold water is permitted, and there is no custom at all to be stringent about this. It is true that from the words of the Or Zarua and the Mordechai it appears that they are stringent even regarding cold water; however, in the source of their words, in the Ra’avyah, it is explained that cold water is in any case permitted. So too testified the gaon Rabbi Israel Isserlein in Responsa Terumat HaDeshen (siman 150), that it seemed to him that in his youth he saw people being lenient to bathe in rivers from Rosh Chodesh Av and onward, and the sages did not object. And so wrote the gaon Rabbi Israel of Bruna in a responsum (siman 12), that many are lenient to enter rivers during these days, and they inferred this from the wording of Maimonides, who prohibited only hot water, but cold water is permitted. The reason is that because of the great heat common in the month of Av, people become covered in sweat, and presumably if they do not wash at least in cold water they will suffer greatly; therefore the custom was not to be stringent in this matter. So too it is testified in the book Shulchan Gavo’ah (siman 551, se’if katan 48), that the custom in Salonika was to swim in the sea, even on the eve of Tisha B’Av, and no one said a word. And the gaon Rabbi Chaim Palagi also testified in his book Ruach Chaim (siman 551, se’if katan 7) regarding the custom in Izmir.

True, there are later authorities who were stringent even regarding cold water, among them in the book Mateh Yehudah and in Ben Ish Chai. Nevertheless, since the entire basis of this stringency is not from the core Jewish law but only by force of custom, and in the Land of Israel there is no clear custom in this matter among the Eastern communities, it seems that one should certainly rule leniently to swim in the sea or in a pool, or to shower in cold water, even during the week in which Tisha B’Av falls. Still, it seems that for our Ashkenazi brothers, who follow the ruling of the Rema, one should be stringent even regarding cold water; only for laborers returning from work drenched in sweat, for whom going without a cold shower would cause great discomfort, may one be lenient and allow them to shower in cold water.
 

However, there is room to be cautious because of the words of the Shulchan Arukh (Orach Chaim 551:18), that one should refrain from things involving danger because of ketev meriri. In these matters I do not understand, although I do not have much confidence in these determinations.