It has always been difficult to get paid for publishing poetry, and only a handful of people can do it for a living. The Internet has not helped the situation and in fact has made it harder to get monetary rewards for writing poetry. However, there are sites that will pay you for your work, as long as it is of the highest quality. These sites will pay you on a poem-by-poem basis, but only if your poems continue to be of high quality.
Write your poetry, and then rewrite it. Never send your poetry off as soon as you have written it, as editors can usually tell when this is the case.
Search for creative writing websites online. Use a search engine to search for online journals, or use a specialised search engine. Duotrope's Digest is a website that searches through online journals in parameters set by you. For instance, you can search only for websites that pay.
Send your poetry to some websites that do not pay. Most paying websites will not even look at your work unless your biography contains a track record of being published, online or in print.
Send your poetry to websites that do pay, after you have built an impressive body of published work. Be sure to send previously unpublished poetry to the new journals, and send as many as they will allow.
Read submission guidelines thoroughly. If you send the wrong number of poems or format them incorrectly, your work will not be accepted. Perseverance is key to getting published; no one gets everything accepted.
Never pay to have your poetry published, online or in print. There are many "vanity publishers" who will print your poetry for a fee, but these are not worth your time or money.