Is Bitcoin an existential threat to Stripe, PayPal’s existing business model? Almost certainly not.
Maybe in the very long term, if Bitcoin were to get near ubiquitous adoption and people regularly used it to transact, instead of preferring fiat (government issued) currencies, then bitcoin might become disruptive to existing payment providers. But even in that case, merchants would still need to be technically savvy enough to accept bitcoin without any help (e.g. the processing services that Stripe and PayPal provide today). A merchant would need to create and manage their own bitcoin wallet to receive funds. And once received, the merchant would then need to be able to use bitcoin to pay for their inventory/labor costs in bitcoin, or would have to handle the conversion to fiat currency (e.g. USD) themselves to be able to pay for things where bitcoin is not accepted.
Far more likely is that Stripe, PayPal and others will continue to provide payment processing services for bitcoin in the same way they provide services for processing other forms of fiat-based payment (e.g. credit cards). And until/unless bitcoin becomes so popular that everyone uses it in place of fiat (note: that will probably never happen), someone will still need to help merchants settle transactions in the merchant’s primary currency (e.g. USD, EUR).
Most merchants who accept bitcoin today do it for the publicity or to serve a very specific demographic of customer. The overwhelming majority of transactions are not completed using bitcoin. In those few transactions that do use bitcoin, the merchant almost always uses someone like Stripe to provide spot conversion from bitcoin to fiat in order to limit currency exchange volatility. For example, if 1 BTC is worth $2500 USD at the time of a transaction, but it’s only worth $2000 when the merchant goes to pay their bills, that’s a 20% loss in value for the merchant. To prevent this kind of volatility, Stripe or others (e.g. Coinbase) provide merchants instant conversion to fiat at the point of transaction, so that 1 BTC gets locked in at $2500 USD and the merchant later settles in USD.
Yet another thing to consider is returns, refunds and dispute resolution. Much of the value from payment methods like Visa and MasterCard comes in the form of well defined rules and requirements for merchants and consumers. If you buy an iPad using your Visa card, and the merchant instead ships you a brick, you (the consumer) are not liable for that fraud. If, however, you had sent that fraudulent merchant bitcoin, you’d be out of luck. Bitcoin transactions are irreversible (at least for now), so the consumer protections and dispute resolution procedures guaranteed by more traditional payment methods (e.g. cards) don’t exist today. Stripe and PayPal (as well as Visa and the issuing banks) play an important role in protecting both merchants and consumers. Even if protections were put in place for bitcoin-based transactions, someone would have to administer them (e.g. Stripe or PayPal) for the merchant; otherwise the merchant and consumer are entirely on their own.
So, in summary, currency exchange, purchase protections, and traditional gateway or payment processing aggregation services will continue to be of value to merchants for the foreseeable future. Stripe, PayPal and others don’t have much to fear from bitcoin any time soon.