The use of "would" in the first part suggests a possibility/probability. The writer can't be certain that criminal activities will go up. It should be noted that the first sentence is rather unnatural for me. "Criminal activities" aren't "more active". The criminals in the area might be more active or criminal activities might be go up/rise.
"give not" is OK. It's rather stylised but I might say something like "I can honestly say I gave not a single moment's thought to what would become of him once we were separated".
Student or Learner