From this GitHub issue: https://github.com/starship/starship/issues/560

Enter this to modify Starship’s configuration:

starship config

Put this near the very top:

# Tuck "new line" inside the ($all) variable
format = """($all
)$character"""

# No new line between shell prompts
add_newline = false

# No line-break between shell prompts
[line_break]
disabled = true

This works because anything inside of ($var) is skipped if $var is empty, and luckily this includes (or does not explicitly exclude/trim) a trailing new-line.