The answer to your question is most likely 01606, which includes the Greendale and Indian Lake neighborhoods. Unfortunately, multi's there are also much more expensive than in other neighborhoods in Worcester, other than 01602, which is very nice, but more expensive than 01606. In the end, it depends on your budget and your tastes. Some neighborhoods in Worcester are under serious pressure from foreclosures, but some neighborhoods have almost none. My parents have lived in the Lennox neighborhood in 01602 for 60 years now, and I can think of 1 foreclosure that has happened in the last two years in that entire neighborhood of 15 streets, and many hundreds of homes.
Worcester is a city of neighborhoods, about fifty in all. Basically, the most expensive neighborhoods are Salisbury, Lennox/Flagg, Tatnuck, West Tatnuck, Midland, Newton Square, and Bancroft, which taken together make up the "West Side." In real estate magazines, the West Side is defined loosely as anything west of Park Avenue, but in character, it is really only about a fifth of the city, geographically speaking, and is made up only of the neighborhoods listed above. Other areas can also be very nice, including Burncoat, Indian Lake, Greendale, and North Worcester.
After the dozen or so neighborhoods listed above, it's really a street by street kind of thing. There are about two dozen or so streets that are worth avoiding entirely, and any real estate agent worth his salt will give you those names and help you to steer clear. Remember, it's a buyers market right now. Don't be afraid to walk away from a deal if it's not a clear bargain. And aim for the best neighborhood you can afford, because if you select one of the neighborhoods stung by the foreclosure blight (none of the neighborhoods in the West Side have seen high foreclosure rates) then recovery is likely to be further off.