I like a straight key to be low to the table. Keys like the J-37/38 may be fairly low designs, but they are usually mounted on some kind of base, which raises the key lever up off the table too far for me. For example I once bought a J-37 key mounted on a piece of Corian sample:
The weight of the Corian helped the stability of the key, but the Corian slab combined with rubber feet underneath raised the height by about 3/4 inch. (The feet were necessary because mounting screws extended beneath the Corian and the Corian itself would have been abrasive to a tabletop.)
One of my favorite straight keys is the "Navy Flameproof Key". Keys of virtually identical design were made by several different manufacturers; mine is by Bunnell. The Flameproof's mechanism is enclosed in a metal housing, which itself is enclosed at the bottom by a metal plate of about 1/8" thickness, that extends forward under the keyer lever slightly. Mounting holes in this plate allow it to be screwed to the tabletop in a permanent installation, or to a base for a movable arrangement. AC2C has a nice photo of a Flameproof key showing this plate, and what the key looks like when mounted to a large base.
I decided to make a very low profile base for my Navy Flameproof key to keep the key knob as close to the tabletop as possible. I removed the key's bottom plate, and installed the rest of the key on a thin but rigid aluminum plate. I cut the plate to a size that I deemed both aesthetically pleasing and stable, and I attached the key using some low profile machine screws inserted from below. Finally, I installed five low-profile plastic stick-on feet to the bottom of the plate. The photos below show the result.
By the way, these photos show my second base design. In the first version, I installed the key so that the key knob extended over the front edge of the plate too far, which made the key unstable: the front feet acted as a pivot point and the rear of the plate would raise off the table. Clearly, ordinary pressure on the key knob created enough leverage to raise up the whole key-base assembly. I fixed this in round 2 by moving the key back as shown.
(Click on photos for large, high resolution versions, in new windows)
 One could also use semi-temporary attachments like Velcro or 3M Dual Lock to hold the key to the table, removably, with its original base. I haven't tried that, partly due the fear of marring the table and a desire to move the key around a bit while sending.