The best way to seal that gap between the spindle and cups (without replacing the whole bottom bracket with a cartridge-type) is to replace the cups with some that have rubber seals built-in, like these:
YST CORPORATION KING RITZ ENTERPRISE CO., LTD. - bicycle parts, threaded head set, ahead-cap, sensor, differential gear device, bicycle accessories range
We used to sell them for less than $10 at my old shop.