As comments on the question already suggest, it looks quite like an address fault when video memory is read. While the picture is not really great, I'd say:
- Blocks of 8 lines are visible, so it must be about A9 of the video address generator (*1).
- Since these 8 lines are displayed twice, A9 must be tied to one level.
- From the content it seems as if the 'second' 8 lines are put over the 'first', A9 must be tied to one (Thanks to tuomas for the correction)
- As continuation across the screen is otherwise undisturbed, the address generator must work as expected, so the fault must be after.
- Since the Mac seems to work otherwise, it can't be on the memory side
Somewhere between the video address generator and the address multiplexer input (joining Video and CPU access) A9 must be tied to ground.
This can be either some faulty trace (have you checked the batteries?), a bad socketed chip, or a 'dead' chip. First step is checking for visible damage and press all chips into their sockets. If still faulty, next step would be to look at the schematics to see if this is still discrete on the SE30 or integrated in a custom chip. If discrete, swapping either the line counter or the multiplexer should solve it, this can be done blind (*2). In case of the later, than you might want to look sad, hold him up, shout "He's dead Jim" and put him over to the spare parts shelf.
I had a short peek at board and schematics, and it seems as if the SE30 is the worst combination possible. While counters and muxes are discrete, they are all SMD devices, so changing them needs soldering on expert level.
The whole logic is rather nice made but different than expected by using 41254 type RAM. These are quite nifty Video RAM that can be accessed like standard 64 Ki by 4 RAM (41464), but offer in addition a 1024 bit line access shifted thru a 4 bit port. The SE30 contains two of these, essentially managing the output of the 4 lines of 512 pixel on its own (well, with a little help of a LS166 8 bit shifter). There is some logic hidden in the PALs used, but it seems straightforward.
This breaks the reasons for fault down to:
either output of VADR(1) on pin 4 (QC) of the LS393 counter at position UF8 being broken,
input C1 (Pin 11) of the F253 mux in position UD8 'broken' (somewhat unlikely)
Both are SMD devices, so not easy to replace.
Or someone pulling the line up to one. A good candidate might be the 16R8B PAL at position UG6 (pin 4 - I3).
This would be great and bad at the same time. Great as it's socketed, bad as I don't know how easy it is to get the image to program a new one (or reprogram if that's possible).
So in either case, you will need to pull out your oscilloscope and start searching in that area.
*1 - SE30 video is 512 by 384 by 1, so 64 byte make a line, 8 lines do 512 byte, so the 10th address line, aka A9, is the one flipping every 8 lines.
*2 - Counter chip first, test again, if still bad, change multiplexer. No test equipment other than the machine needed.