Is working in home decoration with flowers forbidden if it includes decorating for birthdays?

You are not required to avoid working in the marketing of home decorations and flowers, as the default rule is permissibility. This ruling does not change merely due to the possibility of them being used for something forbidden, unless you know that the buyer will acquire them for a forbidden purpose. In that case, it becomes unlawful to deal with them. You are not obliged to question people about their intentions.