I've heard it's not great to use these because it makes the writing seem passive. When would it be okay to use them and how would you be able to keep the writing active?
Thanks!
They're not passive.
They are used either as particples or in the progressive tense.
The progressive tense shows a continuous, ongoing action, rather than an action that has a distinct beginning and end.
He sat at the bar and drank a beer.
That is an action (two actions, actually) begun and completed within that sentence.
But if you say--
When I walked in, he was sitting at the bar drinking a beer.
--then it's clear that he's in the midst of a continuing action, which the POV character did not see the beginning of.
So there are specific uses for the progressive tense. The trick is not to use it when you don't need it.
Then there are participle phrases. These are used in dependent clauses to support the action in the main clause.
She attacked the porch with a broom, sweeping dust and worries into the cracks.
The main action is what she's doing with the broom. The dependent clause that starts with "sweeping" describes how she's doing it. Another important thing to note is that the action in the two parts of the sentence occur simultaneously.
However, sometimes writers come up with clunkers like this (and I've occasionally seen this even in published books):
Dashing up the walk, she rang the doorbell.
Grammatically, what this is doing is taking two equally important, sequential actions and making one a modifier of the other, implying that both happened at the same time, which is physically impossible.* The sentence should be written--
She dashed up the walk and rang the doorbell.
Learning writers seem to be very fond of putting participle phrases at the front of a sentence, to the point where you'll sometimes see four or five in a row. But it can lead to nonsensical sentences, and in general, it's a weaker structure, to be used sparingly. This, for instance, is fine--
Her heart beating wildly, she opened the box.
--but you wouldn't want to use that structure repeatedly. It could also be reversed:
She opened the box, her heart beating wildly -- and still make sense, but you might not want to write it that way, since the most important and tense part of the sentence is the opening of the box, which means it probably should be last.
(*There are some (possibly self-appointed) grammarians who insist this is not so and that it's perfectly fine to write sentences like that. I happen to think they're an abomination (the sentences, not the grammarians). It makes no sense to me to use a subordinate clause as if it were an independent clause.)