Use the Oracle text of a card, not its printed text, to determine that card's wording (C.R. 108.1).
Although Burnt Offering is black, Burnt Offering's Oracle text contains a red mana symbol in its "rules text" (C.R. 202.1, 207.1). Thus, Burnt Offering's color identity includes red and black (C.R. 108.1, 903.4; this is so even though the text of the card as printed in Ice Age has no red mana symbol) and so can't be included in a Commander deck whose commander's color identity is black only (C.R. 903.5c). (See also this thread.)
Although Crypt Ghast is also black and contains (W/B) in its Oracle text's text box, Crypt Ghast contains no colored mana symbols other than black in its mana cost or "rules text" (reminder text is not rules text, and neither are extort's expanded rules [C.R. 207.1, 207.2, 207.2a; review C.R. 702.101a]), so its color identity is black only (C.R. 108.1, 202.1, 207.1, 903.4). Thus, Crypt Ghast, since it's not a card with a basic land type, can be included in a Commander deck whose commander's color identity is black only (C.R. 903.5c-d). (See also this thread and this thread.)
EDIT (May 9, 2023): Edited, including to add opening paragraph and update rule citation.
Burnt Offering
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EDIT (May 9, 2023): Edited, including to add opening paragraph and update rule citation.