In the interpretations of the criterion for the lawfulness of [eating] seafood, legal scholars have said; that which can be derived from hadiths is that the meat of sea creatures should not be eaten, meaning it is unlawful[1] except for the types of fish or sea creature which has scales[2], and crabs do not have them.
In addition to this, there are specific traditions explaining the unlawfulness of crab meat. Imam Musa Ibn Ja’far (Peace be upon him) said: Eating “jari” (a type of fish), and turtles and crabs is haram.[3]
Therefore eating crabs from the point of view of Islam is haram and unlawful.[4]
For more information, see: The Philosophy behind the Lawfulness of Shrimp Question #310.
[1] In other words the primary principal for the lawfulness and unlawfulness of sea creatures is unlawfulness.
[2]Menhajul Mu'mineen (Lil-Marashi), Volume 2, Page 188; Menhajul Salehin (Lil-Khowie), Volume 2, Page 344; Al-Fiqh Alal Mazhahibil Arba'e Wa Mazhabe Ahlel Bait volume 2, Page 20 & Section regarding "That which is forbidden and that which is permissible to eat…" Page 13
[3]Wasa'elul Shia: Volume 24, Page 147
[4]See Jame'ul Masa'el Lil-Bahja: Volume 4, Page 385; Seratul Neja (Tabrizi): Volume 5, Page407