The declaration of Barechu et Hashem Hamevorach (Bless the Lord, who is blessed) and the subsequent response of the kahal (congregation), Baruch Hashem Hamevorach Le'olam Va'ed (Blessed is the blessed Lord forever), belong in the category of prayers known as Devarim Shebikdushah (matters of holiness), among the highest, most intense, and most sacred prayers that we have in our tefillah (prayer), alongside Kaddish (sanctification prayer) and Kedushah (sanctification). And therefore, they require a minyan (prayer quorum). They cannot be said by an individual. Likewise, anytime a person hears Kaddish or Kedushah, they have to pause and respond to the Kaddish and Kedushah. They cannot pretend as if they are not being said. And the same goes for Barechu. If a person came in, let's say, to start a tefillah, and the previous minyan is finishing, and right before Aleinu (it is upon us) they say Barechu, he must respond to that Barechu. Anytime a person hears somebody say Barechu, they cannot ignore it. They have to join in. To ignore it would be to say, “I don't wish to offer a Beracha (blessing) and praise to HaKadosh Baruch Hu.” Even if a person is coming in at the end of the day, ready to pray Mincha (afternoon prayer), and hears the other minyan say Barechu in tefillat Arvit (evening prayer), he still responds to Barechu.
There's only one instance where a person does not respond to Barechu, and that's on Friday afternoon. If a person is coming in on Friday afternoon to pray Mincha and hears the previous minyan say Barechu, which is part of Kabbalat Shabbat (welcoming the Sabbath), he does not answer Barechu there. If he were to answer, he would automatically accept Shabbat and would be unable to then pray tefillat Mincha. He would have to go straight into Arvit and pray double. So if a person hears Barechu on Friday afternoon, he should not respond, and because he doesn't want to accept Shabbat, he has the license to do so. He could go pray Mincha and then say the Barechu, and then tefillat Arvit.