Does the Word "laden" Carry a Negative Connotation?

Laden is neutral. You can see the difference in meaning by looking at the same sentences but with a truly negative word such as burdened:

• The trees were burdened with fruit.

This difference is especially noticeable when the verb is paired with what would usually be a positive noun (like money):

• They are laden with money
  • (This frames it desirably)
• They are burdened by money
  • (This frames it negatively)

Usage-wise, as you note in the question, laden is often used in both positive and negative contexts (and plenty of neutral ones too). In COCA, a basic collocation search for laden with shows “table” as the word that collocates most strongly with the expression, and the contexts for that are positive (“table laden with comfort foods”) or neutral (“table laden with books”). The next strongest collocate is “heavily”, which is used in contexts with all types of connotations such as the positive “heavily laden with the fruit” or the negative “heavily laden with historical baggage”.

Going down the list of collocations, there are some groups that have negative connotations, some with positive connotations, and some that can go either way or are neutral. Either way, neither connotation is much more popular than the other.