An alcohol is more polar than a ketone because it can engage in hydrogen bonding with more molecules of itself or with polar solvents. A ketone can hydrogen bond with other polar molecules with acidic hydrogens, but it cannot hydrogen bond with itself.
Alcohols have a different type of reactivity than ketones, and I'm not certain what you mean when you say that alcohols are more reactive. Ketones tend to react by nucleophilic addition/elimination reactions, while alcohols exhibit other kinds of chemistry, such as eliminations and substitutions.