Q:

Connotation of "appease"

Is "Bob did what he could in his capability to appease them" a positive or negative comment about Bob?

A:

The term appease itself is fairly neutral:

appease - verb
pacify or placate (someone) by acceding to their demands.
assuage or satisfy (a demand or a feeling).

It's not defined as being disparaging, and you can use it fairly neutrally.

I appeased my growling stomach by eating a sandwich.
We appeased the opposing parties, by throwing some concessions in.

However, it does have some negative connotations, going back to pre-World War II, where Britain's policy toward Hitler under Neville Chamberlain was one of appeasement.
This has been heavily criticised as allowing Hitler to gain momentum and power, and World War II happening.
See the Wikipedia article on Appeasement for more details.
See also this question:
What is the polite word describing a person who unreasonably says and does things to make a person happy?

A:

Appeasement as a national policy got negative connotation during WWII:
"The term is most often applied to the foreign policy of the British Prime Minister Neville Chamberlain towards Nazi Germany between 1937 and 1939. His policies of avoiding war with Germany have been the subject of intense debate for seventy years among academics, politicians and diplomats. The historians' assessments have ranged from condemnation for allowing Adolf Hitler's Germany to grow too strong, to the judgment that he had no alternative and acted in Britain's best interests. At the time, these concessions were widely seen as positive, and the Munich Pact concluded on 30 September 1938 among Germany, Britain, France, and Italy prompted Chamberlain to announce that he had secured "peace for our time." (http://en.wikipedia.org/wiki/Appeasement)
Regarding actions of a single human, I'd say the connotations are fairly neutral.

A:

Bob did what he could to appease them.

I would consider them to be the insulted party here. Appease is being used in the sense of soothing or pacifying. Adults don't need to be soothed or pacified.

