If your child, sibling or parent is receiving welfare, that doesn't prevent you claiming them as a dependent. They will have to meet the Internal Revenue Service's (IRS) definition of a dependent, which does include some financial requirements. Those requirements, however, are not affected by the source of your dependent's income, only by the amounts involved.
A dependent must be either a qualifying child or a qualifying relative. A qualifying child can be your child, stepchild, foster child, sibling, step-sibling or half-sibling, or the child or descendant of any of them. She must be either under 19; a full-time student under 24; or permanently, totally disabled. Unless disabled, she must be younger than you or your spouse. She must also live with you for half the year, with exceptions for school, military service or joint custody.
A qualifying relative can be of any age but is not your qualifying child or the qualifying child of anyone else. If the individual lives with you year-round -- except absences for such things as hospital stays or school -- he qualifies even if he's not a blood relative. Some relatives, such as your siblings, parents or adult children, do not have to live with you to qualify if they meet the support test. There's an absolute limit, adjusted each year, to how much taxable income qualifying relatives can have, from any source.
You can only claim someone as a dependent if you provide more than half her support for the year. If she receives Social Security, for example, and the benefits pay more of her upkeep than you contribute, you can't claim her. If she receives welfare benefits but doesn't spend them, the unspent money doesn't count as support. If your support is in a form other than cash -- you let your mother stay rent-free in a garage apartment, for example -- figure the fair-market value of your assistance.
If you help support both your parents, you must qualify them individually as dependents, not as a couple; if only one parent qualifies, you claim one exemption. If you live with your qualifying relative in his house for free, you have to subtract the value of the rent from the support you provide. Money you spend to benefit the entire family -- a new computer-game system that everyone shares, for evidence -- isn't support money, even if one individual uses it a lot.