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 except for the types of fish or sea creature which has scales, 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.
Therefore eating crabs from the point of view of Islam is haram and unlawful.
For more information, see: The Philosophy behind the Lawfulness of Shrimp Question #310.
 In other words the primary principal for the lawfulness and unlawfulness of sea creatures is unlawfulness.
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
Wasa'elul Shia: Volume 24, Page 147
See Jame'ul Masa'el Lil-Bahja: Volume 4, Page 385; Seratul Neja (Tabrizi): Volume 5, Page407