Short answer since I'm drinking beer and on my phone, but I love the question as colonial banking is a particular interest of mine.

The short of it is; it depends on the economic value of the colony- but generally access to lines of credit are more likely to be available the later in the century as colonial economies become developed enough to support more stable banking institutions.

So in the French Pacific/Southeast Asian context, there wouldn't be lines of credit available in the colony until the late 1860s and 70s when the Banc de la Nouvelle Caledonie was opening up branches- though credit was mostly available to people looking to establish residency and chase economic opportunity- not necessarily open to tourists who hadn't made previous arrangements at the BNC offices in Paris. Following the bankruptcy of the BNC in the 1870s there was no easy credit available until the 1890s when the Banc de l'Indochine began a rapid expansion and offered a bit more of the sort of consumer side of banking- they also had branches in France, so tourists from the US or from France were advised to make arrangements with the Paris office before traveling- and I'll have to double check- but I believe they had representatives in San Francisco with whom arrangements could be made, though the point of those representatives was to manage capital/goods coming into the US from the Pacific colonies. There were also local Agricultural and Industrial banks/societies- but they were mostly open only to members and only to pursue economic development- their capital was often provided by the local colonial governments.

Otherwise, even until the 1920s, French tourists and potential settlers to Tahiti for instance were advised to bring cash (either French or American) as banking was not a guarantee- so it doesn't seem that even then banking was necessarily that reliable. In any event, in a 1923 Guide to Tahiti they recommend making contact with pan-imperial banks like the Banque de l'indochine in Paris to see what arrangements could be made, they also recommend bringing notes from the Banque de France which they suggest could be exchanged with the local Chambre de Commerce. An 1899 Guide to the New Hebrides recommends basically the same thing- make arrangements with the Banque de l'Indochine and pick up your money in Noumea (New Caledonia), the last stop, before arriving in the New Hebrides.