The recession seems to have made Americans sick in record numbers, causing serious problems for the Social Security Administration. Since 1989, applications for federal disability benefits have shot up 43%. Benefit claims always rise in slumps because sick or injured workers who might soldier on if jobs were available opt for disability. New regulations issued last year made it easier to qualify, too. And Social Security Commissioner Gwendolyn S. King chalks up part of the increase to an outreach program designed to find eligible recipients who had failed to apply. Trouble is, the sharp rise in claims is draining the disability insurance trust fund at an alarming rate. New projections show that the fund, which has also been hurt by a decline in payroll tax receipts, could run dry by 1997. Meanwhile, Social Security offices are reeling under the caseload--the processing backlog for new disability claims is up to seven months. Congress may appropriate extra money to help Social Security cope and may also authorize a transfer of money from the main retirement trust fund.