I’m not sure where this concern about magnets is coming from, but perhaps a little perspective is in order.
Seriously, for a magnet to damage a computerized sewing machine, we’re talking about something that Wile E. Coyote would have to order from ACME. You wouldn’t have to put it on the machine; the machine — as well as your stove, washing machine, iron, refrigerator, and other appliances — would inexorably move toward it¹. It would affect fluorescent lights and old tube monitors. You would need to post warning signs for people with pacemakers. It, not the sewing machine, would be the most expensive piece of metal you own. If it is a permanent magnet, Chinese rare-earth miners would send New Years gifts to you for supporting their families. If it is a current-driven electromagnet, the board of your local power utility sends you Christmas cards.
The original computer memories were magnetic core memory. These were big 3-D networks of wire and little rings that had to be woven together. The Wikipedia article on this is very good. The lore when I was a freshmen engineering student was that IBM had done a study and found that the best place to make them was Hong Kong, where the women had the most adept fingers from a lifetime of sewing. A very strong magnet might indeed affect one of these, but the likelihood of any sewing machine having one is remote at best.
More recently, like the 1970s-1990s, people transferred files on floppy disks. These were basically made on the same principle as the old analog cassette tape, using a magnetic field and magnetic head. Magnets could definitely ruin one of these.
Most recently, though, files are transferred by Ethernet and memory stick and stored on hard drives. These should not be affected by the relatively weak fields you can generate with a magnet that is essentially an overgrown refrigerator magnet. I can’t speak to all of the possible computerized sewing machines out there, but very few if any of them will have hard disk drives, so that is not an issue. Even if they did, the drives themselves are encased in a metal container that shields the normal magnetic fields one is likely to encounter inside the computer. It is worth pointing out that the computer power supply itself has an enormous transformer that creates a fairly powerful (albeit mostly self-contained) magnetic field, and that every current-conducting wire generates a magnetic field – every one of them! Look inside your computer: there’s the power supply, a bunch of wires coming off of it whose purpose is to conduct current, the memory modules are sitting there unshielded on top of the mother board, and the hard drive is inside the case with all of this. Do you really suppose that if “magnets” were a concern for memory or hard drives, they would design the computer that way? Also, please see this article, in which PC World busts the myth that anything less than a magnet designed for the purpose could erase your hard drive.
That leads me to what is most likely inside the computer control for the sewing machine. This particular button hole machine does have a fancy interface to control the settings, which it remembers. That implies that there is a computer with a memory that persists after power-down. That computer is mostly likely a micro-controller running off of some kind of permanently hard-coded memory (or at least memory that requires special methods to change) that is not susceptible to magnetic fields, and the settings are stored on memory that is similarly resistant to magnetic fields.
As Grace suggests, the field should drop off rapidly with distance from the magnet. R^2 is the square of the distance, and the field falls off inversely proportional to this. At a point twice as far away, the field is 1/4; at a point 3 times as far away, 1/9. This is true if the magnet is in free space. However, if the magnet is stuck to a nice piece of steel, the field falls off even more rapidly in the direction of the steel. Given the nice thick steel plates that makes up these industrial machines, the magnetic field is not likely to permeate to the poor little computer that is, incidentally, underneath the table, all the way in the back. That’s like 3′ away through multiple steel plates. Generally speaking, you are not able to lift the magnet that would be capable of affecting the computer through that much shielding.
¹ At 4:28 minutes begins the magnet segment where you can see everything from cars, buses, ocean liners, the Eiffel Tower and even satellites pulled into the magnet. Also, please note the author of this piece is Mr. Fashion-Incubator who is an electrical engineer ~kf