The red tulips that Serena Joy is trying to grow in her garden represent life and fertility -- the key theme of the novel. Serena Joy is a wife, but she can't bear her own child. Because she can't "grow" her own children she tends her garden. The red of the tulips is certainly suggestive of blood and life. What is notable about her garden though is that even that is failing. Serena works diligently, but the garden never flourishes, just as her womb can't flourish. The flowers themselves look like receptacles, and that is another connection of them as symbols of the women, specifically the handmaids that are the "wombs" of the society -- bearing the children for the wives and the commanders. Serena Joy is subjected to the "ceremony" in a similar, and yet very different way from Offred. Everything in her life depends on Offred's success in getting pregnant, and Serena Joy has little to no control over that, so she takes control of her garden and does what she can there, event though it appears that the garden will fail as well.
Everything in regards to the Handmaid is relative to red. In the book The Handmaid's Tale the world as people know it has changed. Inside the country women are nothing and men are in control. The fanatics have revised life so that people live in status roles. Infertility is rampant and the objective is to try and reproduce children into the households of people of authority.
The handmaids are a select group of women who are imprisoned by the society to serve as pregnancy vessels. They are forced to wear red as their attire. The color red symbolizes life's blood. Therefore, the red tulip symbolizes the life blood which nourishes the infant as well.