べき

Appropriate to/Best to
Verb Dictionary form + べき
な Adjectives/Nouns: + であるべき
い Adjectives: く form ＋ あるべき
する→すべき

べき describes the appropriate or best course of action.
Weblio defines べき as describing “Obvious“, “Appropriate“, or “Obligated” courses of action. 
It is usually translated as “I/You/He/Should…”
Its meaning is most similar to しなければならない and たほうがいい
Please be careful not to confuse this pattern with はず (a different kind of should)

べき is an auxiliary verb and can directly modify nouns (Like an い adjective) and uses だ like nouns. 
べき tends to use the ”Aはべき” pattern, but Aを・にべき is ok as well. (topic/object difference)

Examples

• ご飯は全部食べるべきです。- Food should be completely eaten.
• ご飯を全部食べるべきです – You should eat all of your food. 
• 酒はほどほどに飲むべきです。You should drink in moderation.
• 自分のミスはしっかり責任を取るべきだ。- You should take responsibility for your mistakes.
• テストの前は勉強すべきだ。- You should study before a test.
• 親の言うことは聞くべきだった。- You should have listened to your parents.
• 今日にすべき仕事だ。－ Work that should be done today. (That you should do)
• 謝るべきだけど、やっぱりできない – I know I should apologize, but I just can’t do it. 

べきではない

Inappropriate Course of Action/Not good to
べき form + ではない

While べきじゃない is kind of acceptable, the negative of べき is almost always going to be べきではない (See below).

• 人は殴るべきではない – People shouldn’t be hit.
• 人を殴るべきではない – You shouldn’t hit people. 
• 嘘はつくべきではない – You shouldn’t lie.
• すべきではないことをしてしまった。- Did something that shouldn’t have been done.
• 仕事をせず、一日中酒を飲むべきではない – You shouldn’t spend all day drinking without working.
   

ないべき

Some young people are also apparently using verb ない form ＋ べき to say this, but I wouldn’t recommend it. Its a new development and sounds a bit uneducated (to me). Here‘s a Japanese forum post on the issue (Primarily older people) talking about how this is wrong and feels bad. 

You are also probably not going to see ないべき on the JLPT, but べきではない will definitely appear on the test. 

すべき or するべき

To cut a long story short, use すべき。

A survey by the NHK found that most people generally prefer すべき。(depending on the construction, 55 to 62% saying you can only use すべき or preferring it). 

This is probably because, べき is a more literary word (Written), coming from a much older Japanese word. It is still spoken, but because of this literary connection, it is usually preferred to use the dictionary form of する which is す. It probably also because of this that べきではない is far more common than べきじゃない。

NHK Article on the issue

べき and はず

To quickly compare the two:
はず – Probably did, Was supposed to have done, Likely to do/Likely to be
べき – Should be done, Would be best to do, Best course of action is…/Appropriate action is