Genoa is a relatively safe city for tourists, as is the case with the entire Italy. During the day, the only inconvenience that may arise is pickpockets (there are many of them in squares, viewpoints, and other crowded tourist areas).
As darkness falls, the situation changes: walks alone, especially for girls, should be cancelled. Or take a taxi - for single women in Genoa there is a 10% discount on taxis from 10pm to 4am (although taxi drivers often "forget" about it - be sure to remind them!).
In some coastal areas of the city (especially those that run parallel to Via Gramschi) it's best not to go at all unless it's absolutely necessary.
