Vaccination is recommended for all persons, including breastfeeding women, who are contacts of infants or children aged 59 months and younger (i.e., <5 years), because infants and young children are at high risk for influenza complications and are more likely to require medical care or hospitalization if infected. Breastfeeding does not affect the immune response adversely and is not a contraindication for vaccination. Women who are breastfeeding can receive either TIV or LAIV unless contraindicated because of other medical conditions.
https://www.cdc.gov/flu/professionals/acip/specificpopulations.htm#breastfeeding
This is from the CDC website.