Apart from making your way round the Caribbean or Polynesia by offering a hand on board yachts which seems to be common and easy, the most common route is the Atlantic crossing from Europe to the Americas (mostly to the Caribbean or Brazil). See Cruising on small craft for more general information on yachts.
There are stories floating around of people who always know someone else who managed to go for free on a freighter, but the only reliable stories really date back to the 1970s. Cargo ship travelling is commercialised now virtually everywhere. For something like per day you can rent a cabin on them.
Be assured that once you hit the marinas everything will be pretty obvious. You will meet other boat hitchhikers and they will share their information with you. Basically you'll be putting up notices offering your help, pacing the docks, approaching people cleaning their yachts, trying to make contact with sailors in the bar, etc. Try to talk to as many people as possible. After a while everyone will know you and will give you hints as to which boat is looking for someone.
If you prefer more high-tech methods, there are several on-line marine crew websites available that specialise in matching crew with boats:
Experience: Experience in sailing is not necessary - although a huge plus in getting a quicker lift - but participating in duties and life on board, of course, is obligatory!
Some money: These days most yachts will ask you to chip in for your food. Most boats ask either €10 or €15 on food per day. The Atlantic crossing takes between 15 to 25 days, depending on boat and winds, so you'll need around €150-300 for the crossing alone - count on at least a week in the marina till you find your lift, too. It is possible to get an entirely free lift food included, mostly on bigger yachts where you'll be needed to scrub the deck and polish the silver railing. During the ARC regatta you can even expect boats ask as much as €50 per day since inscription fees are pretty high.
Things you'll need: Waterproofs and good shoes are recommended although they are very expensive and if you don’t have them you can wait -the boat you get might provide them and only if not you'll have to acquire them. The Canary Islands are cheaper when compared with mainland Europe and in Gibraltar there are sailor’s “jumble sales”, so you might pay less.
Language: Obviously all foreign languages can help, but for the Atlantic crossing speaking at least some French can make a huge difference, since over 60% of sailors on this route are French-speakers.
Boats go with the trade winds that start to move from East to West across the Atlantic in autumn, so the season is from September to January or February. Top season is as from November. In the end of November each year there is a regatta called ARC (Atlantic Rally for Cruisers) leaving from Las Palmas. There will be more boats than at any other time in the marinas and it can be considered safer than leaving with a boat that is going unassisted. There will be more competition on finding a lift though, too.
Winds have become less predictable, and more storms have been happening because of global climate change. The way most sailors comment on this is "the winds are going through a transitional phase to find a new rhythm".
If you don’t want to pay the ferry in Algeciras it is possible to sail to Africa, although unless you are extraordinarily lucky, you’ll have to go via the Cape Verde islands which are a stopover for many trans-atlantic sailors. They are between 10 and 14 days from Gibraltar. From there you’ll have to catch a new boat to get to Senegal which is three days away. It is an experience in itself although maybe not the perfect swap for a one-and-a-half hour long ferry ride that’ll cost you €25. You also miss out on Morocco and the crossing of the Sahara Desert, which are highlights of any visit to Africa.
Ocean Nomad: The Complete Atlantic Sailing Crew Guide - How to Catch a Ride and Make a Difference for a Healthier Ocean
The Pacific being about 6 times larger than the Atlantic, this is a crossing that is never done without a stop over on at least one of the Polynesian Islands. The most likely end-point for most is Australia or New Zealand.
Boats begin to leave the west coast of Mexico for French Polynesia in March and continue to sail as late as June and July. The cyclone season, or off season, in the South Pacific begins in November. The initial crossing from Mexico to French Polynesia is roughly 2,500 miles and will take 3–4 weeks on an average sized sailing vessel that doesn't carry much fuel. Times very depending on the length of time a boat is stuck in the doldrums or inter-tropical convergence zone (ITCZ).
Passages from Australia to Asia can be made with the Sail Indonesia Rally, which takes place annually in July. The other way round it is more difficult. You might be able to join a returning yacht, but many do not return straight away, so you may have difficulties finding them.
Transits will be timed to the winds, with the main obstacle being the strong south-eastern trade wind (especially for those wanting to continue down the Queensland coast). Regarding this the best time for a passage from Bali to Darwin is Christmas to March (it is possible to find the ideal crossing time using the beautiful rendered world wind map) – but that is hurricane season, so an earlier departure may be chosen.
The cost of the passage will depend on the captain. Doubtless you will be asked to contribute for food and perhaps fuel (entirely dependent on the captain. As for subscribing to the websites above (findacrew is the only subscription), this should not be necessary as a serious captain will buy this to advertise himself.
Ideal places to find captains in search of crew are: