When you experience Jamaican culture you will see that the Jamaican people are very generous and warm. Returning this warmth and friendliness is a great way to show them you appreciate their country.
Chances are, you will be approached at one point or another during your travels in Jamaica for money. Do not feel pressured into giving money. A strong "I'm alright" and walking away is usually the best advice for instances such as this.
This also applies in the infamous straw markets. Note that the European method of just walking away does not work well. You will generally need to engage with someone in order to get away from them.
That being said, if you befriend or encounter one of the many wonderful Jamaican people and you wish to give a friendly gift, that is perfectly acceptable and welcome. Just exercise common sense when it comes to money.
Cultural respect is far more important. You are guests on their island. Please know also that when speaking to the elderly you should say, "Yes ma'am." or "Yes, sir". Good manners should be displayed at all times. Respect the environment and the people.
It is a simple rule of thumb that should always be applied when travelling abroad. Don't expect that everyone will respect you, however.