Many of the answers seem to agree that because you would be using TLS you can rest assured. There seems to be some disagreement between whether landline, wired Internet or satellite Internet is safest.
It's important to mention that TLS has had its share of problems and that TLS interception is possible. The other issue is that a connection from a private residence may or may not be of interest to someone but it's also a question of accessibility and reward for the effort.
User MooseBoys touched on this in a comment to the question:
"It's worth pointing out that the nuclear arsenal of the United States is controlled, in part, via satellite connections.".
User Lie Ryan also brushed against the problem in their answer:
"It's likely that your satellite service provider might have already have an over-the-air encryption setup on their network. I can't fathom a commercial satellite internet provider that would be so irresponsible as to not use any form of over the air encryption. Like VPN, this would move the practical interception points to the ground based gateways, though that gateway will be one controlled by the satellite internet provider rather than an independent service.".
Attack on a country's phone and Internet is a useful military objective but at least part and usually all of the most secure information uses different circuits - I don't want to travel too far down that path because your banking information isn't going that way and you are asking about satellites.
While attack on a country's internal communications is a useful objective for an aggressor to perfect and an active field of study it is space where control is important. Underwater cables (where all landlines travel country to country) have common and fixed entry and exit points, the country where they enter (for forwarding throughout the continent) has physical access to them and prior attacks on them have been documented.
L3Harris advertises equipment to intercept encrypted satellite communications, see their webpage: "Satellite Intercept":
"The proliferation in organised crime, and its interaction with global terrorism, presents challenges for naval, marine, police and security forces. Monitoring, tracking and taking measures to stop the actions of these groups costs both time and money. So ensuring that actions are carried out effectively and efficiently is crucial.
The relative ease of communication between individuals and groups, due to the extensive coverage and availability of Thuraya and Inmarsat satellite networks, means that they are favoured by criminal and subversive groups located in remote or even urban regions.
The ability to intercept and monitor conversations and data transmitted over satellite networks, provides the vital Intelligence which enables detailed and comprehensive analysis and reporting of target activities. This gives the operational commander the ability to search for, intercept, identify and locate potential targets through EW assets, to ultimately win the information battle.".
It is alleged that the Russian satellites Olymp-K (Russian: Олимп-К meaning Olympus) and Luch (Russian: Луч meaning ray and sometimes transliterated as Loutch) have been involved in satellite espionage.
Stratign is another company that advertises satellite intercept systems:
"Satellite Turnkey Interception System
In today’s challenging security environment, Governments need to monitor conversations, fax and email messages and other data transmitted across satellite networks for national and international communications. Whilst most satellite communications are perfectly legitimate, the extensive geographical coverage provided by satellite communications networks means they are favored by subversive groups in remote regions. Thus, Satellite interception system is a basic requirement for countrywide intelligence collection operations.
The Satellite Interception System will provide an automated platform for the Intelligence agencies to conduct passive collection of satellite transported information and achieve plain Voice/ fax and data information. In case the transported information is encrypted, the Satellite Interception system will provide encrypted raw output. Decryption is not part of this system and should be treated as a separate activity altogether.
Satellite Interception System provides its users with a fast Satellite data acquisition and Online Demodulation. The system has a Carrier Interception application integrated for finding new targets and achieving all necessary measurements. The system has been adapted in a manner to perform online DCME Analysis for current and new DCME systems being adapted by various subscribers globally.
Once the Satellite interception system is tuned to any of the satellite uplink and downlink frequency, all satellite information transported by these countries will be present for the interception. The user can then, based on his experience of using the carrier interception system, has to select the channel of the country of interest and tune the receiver to receive that channel of interest. The system will automatically demodulate the satellite signal and adapt the decompression scheme for extracting Voice/Fax and Data outputs for the intelligence operatives to view and act in accordance with the requirements.".
Day to day you shouldn't worry about your banking information or other secured transmissions.
Question: Is it safe to pay bills and access accounts through this? Specifically, should we be be concerned that someone can just snag our signal out of the air and steal our information? I know anything is possible, but is it likely?
A foreign government probably does "steal your information" and almost certainly does nothing with it, so as not to reveal themselves and their success. One day, if there is war, they might send send transactions from many accounts along with interfering with other transmissions in order to bring electronic chaos to the threatened country.
This really isn't something to worry about on a day to day basis, and you are insured against such fraud. Privacy is another issue, do you want what you purchase or which websites you visit to be known, again this probably won't be an issue unless you are famous; and especially if you are a world leader.