How to pronounce "p" in "hospital" and why?

Solution 1:

When the phoneme /p/ occurs at the start of a stressed syllable in English, it is aspirated with a little puff of air, making it a [pʰ]. Contrast hospital with a [p] with husband with a [b]. Hear the difference now? It is subtle, but you can have a loss of aspiration without having to gain voicing at the same time. But people mistake this in hearing.

Per the Wikipedia article on English Phonology:

In most dialects, the fortis stops and affricate /p, t, tʃ, k/ have many different allophones, and are distinguished from the lenis stops and affricate /b, d, dʒ, g/ by several phonetic features. They may be aspirated [pʰ], voiceless unaspirated or tenuis [p⁼], preglottalized [ʔp or [ˀp], or unreleased [p̚].

• Fortis stops /p, t, k/ are aspirated [pʰ, tʰ, kʰ] when they occur at the beginning of a word, as in tomato, trip, or at the beginning of a stressed syllable in the middle of a word, as in potato. They are unaspirated [p, t, k] after /s/, as in stan, span, scan, and at the ends of syllables, as in mat, map, mac.

Because the p in hospital occurs after an s, it cannot be aspirated. Since a /b/ is never aspirated in English, you are misperceiving the loss of aspiration as a signal that you have gone from [p] to [b], but that is not what is happening. Unaspirated [p] is still unvoiced, whereas [b] is voiced.

As Professor Lawler observes in comments, it can be hard to hear the difference in these contexts. Consider the common error of spelling disbursement as *dispursement. There is not much to go on there.

So your p in hospital is still a valid allophone of /p/: [p] not [b].

Solution 2:

P turns into B very easily, since the difference between them is just unvoiced versus voiced. Folks may substitute one for the other without realizing they've done so... including people like myself who pronounce it as P when we're paying attention

My advice would be to continue to use the P but not worry about it. You'll learn how sloppy you can be and still be understood... but don't deliberately start out sloppy.

(There may also be regional accent effects. One is cited in the comments. Another: my own accent is a mixture of New York and Boston, and when I'm not being careful I'm likely to pronounce it as "hozpital" -- voicing the s, not the p.)