Learn Dutch: Je kunt de dozen beter sluiten door veel plakband te gebruiken. - You'd better close the boxes by using plenty of adhesive tape.

In Dutch, combining the verb kunnen (to be able to) with beter (better) is the standard, natural way to give advice, translating directly to "you had better" or "you should" in English. Literally, you are saying "you can better close the boxes".

How does the door ... te gebruiken construction work here?

The pattern door + [objects/details] + te + infinitive is used to mean "by doing [something]". In this sentence, door veel plakband te gebruiken translates to "by using plenty of tape". Notice that the object (veel plakband) gets sandwiched between door and the infinitive phrase at the end.

If I wanted to estimate and say "by using about three rolls of tape", how would I add the een stuk of structure to this sentence?

You would say door een stuk of drie rollen plakband te gebruiken. Here, een stuk of means "about" or "around". It precedes the number (drie) and the unit (rollen), which is then followed by the substance (plakband).

Could I say vele plakband instead of veel plakband?

No, because plakband (adhesive tape) is an uncountable noun. Just like in English where you say "much tape" rather than "many tapes", Dutch uses veel for uncountable mass nouns. Vele is generally reserved for plural, countable nouns.