Correct. Surprisingly, both the students and the lecturers considered the strike to be necessary.
To be fair, the mutual support wasn’t surprising; the students could become lecturers too someday and, if if they don’t see that as their career path, they recognise the value of the lecturers!
The common issue with using “both” is that it actually adds ambiguity… if you say “both the students and lecturers” it is unclear if you mean all the students or just two students (both the students).
Note that the phrase “Surprisingly, the studens and lecturers…” would have been totally fine, so my advice is to always ask yourself do you really need to use both at all? If you really want the extra emphasis that using “both” can add, then make sure you use the same construction for each of the groups; “the students” should be matched with “the lecturers”.
That or Which