What I do in my serious self-sufficient forts is similar to Sphalerite's idea, but a bit more complicated.
Each kind of crop I want to grow gets its own farm plot. I like 4x5 plots for this, because 20 is a goodly number while still being small enough for even the most easily distracted planter to service in a timely fashion. I usually end up with about ten plots total, four or five underground ones (depending on whether I'm growing quarry bushes) and one for each above-ground crop I've plant-gathered or traded for seeds for. Each plot will be set to grow one season only, either spring or summer. I try to kinda load-balance my post-processing jobs, by growing rope reeds (for cloth) in the spring and pig tails in the summer, cave wheat (for flour) in the summer and whip vines in the spring, that kinda thing. Point is, each plot is only planted one season out of four.
Next to each plot is a 4x5 stockpile — same size as the plot — set to accept only that plant, with the number of barrels set to zero. When it's harvesting time, my dwarves just pick up the plants from the plot and drop them in the stockpile right next door.
Now, this may not actually be true … but I believe plants in stockpiles, even not in barrels, are effectively in suspended animation. They never rot. At least, they've never rotted for me. Which means when each plant stockpile fills up, the dwarves will keep planting, but just let the plants rot in the fields and then take them to the refuse pile. This continues until I run out of each particular type of seeds, at which point they simply stop planting that crop.
But that's okay, because I've got seeds sitting there in the plant stockpiles, preserved in plant form. As soon as I order something to be brewed or processed or milled (or the dwarves grab a strawberry or plump helmet for a snack or something) I get the stored seeds back. That leaves holes in the plant stockpiles, which get filled at the next harvest, and we're back to the status quo again.
Now, I believe the game limits how many of each type of seed you have; you get a maximum of 200 of each type of seed, and once you hit that limit, things that should produce seeds (like brewing, milling or eating berries or mushrooms raw) just stop doing so. Seeds are stored in bags, and each bag holds 100 seeds, meaning you will at most have two bags of each type of seed. If you want to be really cautious, you can just wait for your dwarves to collect one full bag of some type of seed and start filling another, then forbid the first, full bag. Sometimes just for flavor I have them cart one full bag of each type of seed down into a locked room; that's my seed vault. As long as those seeds are safe, I know even if I flood my greenhouses with redwater I'll still be able to restart my farms. But I've never needed to use that contingency plan. I really only do it because it amuses me, same way I'll sometimes mint up a few stacks of gold coins and then lock them away in a trapped room. Serves no purpose whatsoever in the game, but it tickles me to do it.