Some old Operating Systems I hear have only BASIC Programming Language. Are there any active and old operating systems which includes only BASIC Programming Language?
(Preface, as I understand from comments, the OP wants to install some OS on a modern machine that has BASIC as main user interface)
Simple Answer: Then there is none.
All OS for modern (PC-class or other) machines are stand alone Systems with some kind of UI (graphics or text). Here, if at all, BASIC is only a language to be loaded via the builtin UI.
As Maury Markowitz already explained well, ye old days software stack differed quite a lot from what we have today. Back then, there where only marginally separated layers and making BASIC language and OS in one was kind of a shortcut to get it working (and within the limits of back then (affordable) hardware).
Having said that, you may of course assign some console based BASIC as shell (*1). Even a classic emulation like Sinclair BASIC might work well. But you also will soon notice that most features of the fine new computer will be inaccessible, and maybe only 48 KiB of the 8 GiB installed are at your fingertip ... but boy they will be fast :))
Essentially this is no difference from just invoking the BASIC/emulation from a regular Linux shell. So no real gain at all - except maybe for fooling friends when logging on ... then again, how many remote users are there on your PC anyway.
Now, if, instead, your intention is to work with BASIC on today's operating systems (as application), then you want for sure take a look at BBC BASIC. Yes, the fine 1980s BASIC is not only still alive and actively maintained by Mr Russell, but has been widely enhanced and ported to many (*2) modern OSes. Up to an extension that would allow to control most features of modern OS again.
Most definitely worth to invest time. So go and install it on top of whatever OS you prefer or there is and enjoy it as single frontend.
*1 - Some heavy configuration fiddling required on modern Linux to bypass login.
*2 - Personally I would even say all 'relevant', but that's subjective, even as the list of maintained ports includes Windows(Vista, 7, 8, 10), Linux, MacOS(OSX), iOS and Android for the SDL branch, as well as MS-DOS, Windows 3.1 ... 10, CP/M, RISC OS, *BSD, Amiga OS** and many others in other branches. Them being widely compatible and fast (depending on the machine) it may even be more of a cross platform deal than C :))
Some old Operating Systems I hear have only BASIC Programming Language
It was more the case that BASIC was the operating system. Various commands for working with devices like floppy disks and printers were added to the dialect of BASIC running on that machine. Turning it on would result in the BASIC READY prompt where you could type in a program, or begin operating the devices.
Perhaps the best example of this was the Commodore line, which had a fair variety of device operating commands in the OS. Other machines, like the Atari, used a combination of commands in the language as well as a separate program (called DOS) for doing less common things like copying files or formatting disks. Most machines of the early 80s varied between these extremes.
RISC OS Pico on the Raspberry Pi boots to BASIC. See https://www.riscosopen.org/content/downloads/raspberry-pi
The "full fat" RISC OS could also be made to boot to BASIC using *CONFIGURE commands like you can with the Archimedes and RISC PC.
Note that riscosopen.org do not list the RasPi4 as being compatible (and RISC OS Pico has a smaller compatibility list than the "full fat" version).
The question is rather unclear, and this answer is beyond retrocomputing. Well...
The ESP32 microcontroller is a current (built in ROM claims 2016) chip that has no operating system as we understand the term. You are supposed to program it on bare hardware (but it has wifi and TCP/IP stack). But, it comes with BASIC included, even though undocumented.
The Parallax Basic Stamp is a system-on-board (wide DIP form-factor) used for education. The only programming language that it supports is BASIC. Code is entered in an IDE on a PC then tokenized and loaded onto the board. An interpreter there runs the bytecode. This is pretty similar to how many micros from the '80s did things. The CPUs used on the Basic Stamp boards are all PICs IIRC but there is no way to run native code on them; the only means of programming is through BASIC (there are no tokenizers for other languages).
There is SpecBas, which is a hybrid between an emulator and an enhanced version of Sinclair Spectrum BASIC.
It does require a host OS to run, though, so it does not entirely fit your bill.
Acorn's original RISC OS for the Archimedes is still alive, maintained, and runs on a variety of modern ARM hardware such as the Raspberry Pi and BeagleBoard range. It includes a lightning-fast BBC BASIC V interpreter as its primary programming language, although it is not written in BBC BASIC itself. (Although parts of it, and the ARM processor, were prototyped in BBC BASIC!)
It only runs on ARM hardware, though. To run it on an x86 machine, you need an emulator, such as RPCEmu (FOSS) http://www.marutan.net/rpcemu/index.php or Virtual Acorn (commercial) http://www.virtualacorn.co.uk/
If you cast your net slightly wider, for instance to the Pascal family, then there is Native Oberon, which is a native PC operating system written entirely in the Oberon programming language. Not a trace of C or anything else. It's tiny, fast, efficient and designed to be easy to learn, understand and modify. http://ignorethecode.net/blog/2009/04/22/oberon/
** UPDATE **
I've found out about 2 new contenders that I did not know about when I posted my previous answer back in January.
The BASIC Engine fits your a bill: a very inexpensive, DIY 32-bit computer with a rich, powerful BASIC: https://basicengine.org/
And the MaxiMite: http://geoffg.net/Maximite_Story.html
(RSTS-11 became RSTS/E with support for other programming languages and with runtime systems providing partial compatibility with other PDP-11 operating systems, but originally it was BASIC-only)
To the best of my knowledge, a single "program" (actually the resident runtime system) provided command interpreter and BASIC-PLUS compilation. But my experience is with RSTS/E, not RSTS-11.
The Pick operating system is mostly accessed through its dialect of BASIC, which was adopted in 1973. Nowadays you can run it as a guest OS on Windows.
The Pick operating system consists of a database, dictionary, query language, procedural language (PROC), peripheral management, multi-user management and a compiled BASIC Programming language