Although it partly depends on when the opponent bursts during your combo or attack string, some characters have variations of combos that allow them to maximize the chance of blocking a blue burst without dropping the combo.
In the video you linked to, for example, Slayer's j.P and j.K just happen to be really burst-safe.
in general, i guess the only "sure" way to react to the burst is by RCing and holding back. that's not to say you SHOULD do that, though -- actually, if you notice the burst before it hits and you can jump-cancel whatever attack you threw out, then go for that, because air blocking has very little guard stun and you might be able to recover fast enough to punish the opponent. and depending on the situation, or even what character you're using, it may not be worthwhile to blow tension on reacting to the burst.
it's the same idea, more or less, for blocking DAAs -- if you're pressuring someone, some attack strings are more "DAA-safe" than others, because of the amount of recovery on the moves. If you can react to it, then either jump-cancel and air guard the DAA -> punish, or RC mid-attack if you feel it's worthwhile. (And of course, just like with blue bursts, it also depends on when the opponent DAAs.)