There are two baptismal sites on the Jordan river, Yardenit and Qasr el Yahud, the former is located fully in Israeli territory and both banks of the river are Israeli. The later, Qasr el Yahud, is the place where, according to Christian tradition, John the Baptist performed the Baptism of Jesus and where, according to Jewish tradition, the Israelites crossed the Jordan River, and the Prophet Elijah ascended to heaven. This is also where the picture in the question was taken at.
The Israeli side of Qasr el Yahud is open to the public since 2011, and the Jordanian side has been open for a lot longer. While it's physically possible to walk from one bank of the river to the other, due to the river being shallow (~1 meter) and narrow (~5 meters) with slow water flow, at the place, the International border between Israel and Jordan passes in the river itself, so it's not allowed and border guards are located on both banks to make sure that it doesn't happen. Do not, under any circumstances try to get across the river there.
If you want to get to the other side, you need to go through a border crossing, the closest one is the Allenby Crossing / King Hussein Bridge. It isn't open to Israeli or Jordanian citizens, only to tourists, West Bank Palestinians and Eastern Jerusalem residents.
You can't get a Jordanian Visa in the Allenby Crossing and need to have it in advance:
Can I get a Visa for Jordan at the Allenby Border Crossing into Jordan?
Please note that you are NOT able to get a visa to Jordan at King Hussein/ Allenby Bridge. You must have a visa in advance to enter Jordan at King Hussein/ Allenby Bridge. You may apply for a visa in your home country, at the Jordanian Embassy in Tel Aviv or the Jordanian Representative Office in Ramallah. However, if you have already entered Jordan and got a visa, [your] visa may still be valid to re-enter
Take notice that the Qasr el Yahud site is located on a border area that is still heavily mined, if you come alone, without a guide, make sure to not go across any fences (they are there to fence minefields) or stray off the road, this includes the abandoned churches at the place which are located inside minefields The site is maintained by the Israeli Nature and Parks Authority and has their representatives on site, if you have any questions you'll be able to ask them.
The Israeli ministry of defence has annunced recently (early December 2018) that the mines have been cleared from 3 of the seven monasteries on the site (Greek-Orthodox, Ethiopian and Franciscan) and that in the coming months control of them will be passed to the respective churches and they will be opened to the public.