Boats suitable for ocean crossings are much more dependant on the skipper, than the actual boat design. Size is the obvious limitation, for a few reasons, the first being able to carry enough of what you need, like water, food and enough speed to get you there in the allotted time. You can cross in a 20' Flica, but you'll be packed to the gills to carry enough the 3 weeks of food and water to make a 3,000 passage. That's a long time to spend aboard with barely enough room to lie down, though plenty have done it, most find this a bit cramped.
At 25', boats have enough elbow room, storage, stores volume and motion, to make the trip much more tolerable. 30' - 35' seems a very common cruiser size, being small enough to still easily handle solo, pay for slip fees, while still offering comfort, stores room, etc.
As far as a low cost design, well there's no such thing. Building new wouldn't be the economical route to swim. Buying a well used 26' - 29' something or other, fitting her out as you want her, then casting off would be the most economical way, to get it done. I know of several boats you can have for a song. A Cal 40 is available near here, on a mooring, sailed occasionally, needs some work, but is currently a functioning sailboat for less than $10k. She's not ready for an Atlantic crossing, but some sails, winches, rigging, general repairs, etc. and she would be.