As in the title, which of the two is better:
- During the last economic crisis, many families lived from hand to mouth.
- During the last economic crisis, many families were living from hand to mouth.
English style guides recommend not using the continuous tense unless it's integral to the meaning of the sentence. This is because certain words in a sentence can detract from the readers sense that the writer is certain about the subject -- every little word introduces some measure of doubt. Someone who is sure of themselves uses short, declarative sentences. Compare these:
She eats muesli for breakfast every morning.
She is eating muesli for breakfast every morning.
The first is clear, simple, and certain. The second introduces a slight uncertainty, as if the author is not sure if her breakfast is the same every morning.
That being said, these style guides are for formal, expository writing. With creative writing you say whatever sounds best to you. Also, as you get more comfortable with expository writing, you define your own style.
In your example there is actually a slight difference in nuance between the simple past and the simple past continuous tense. The first says "this happened". The second says "this was happening (while something else was going on)".
If you plan to talk about some other contemporaneous event -- which is strongly implied by your use of "during" -- then the continuous tense is appropriate. Otherwise, your default should be the simple past.
While he is watching the pitcher, a good baseball player will also note the movements of every other player on the field.