I like the WMIC and TASKLIST tools but they are not available in home/basic editions of windows.Another way is to use QPROCESS command available on almost every windows machine  (for the ones that have terminal services - I think only win XP without SP2 , so practialy every windows machine):
@echo off
:check_process
setlocal
if "%~1" equ "" echo pass the process name as forst argument && exit /b 1
:: first argument is the process you want to check if running
set process_to_check=%~1
:: QPROCESS can display only the first 12 symbols of the running process
:: If other tool is used the line bellow could be deleted
set process_to_check=%process_to_check:~0,12%
QPROCESS * | find /i "%process_to_check%" >nul 2>&1 && (
    echo process %process_to_check%  is running
) || (
    echo process %process_to_check%  is not running
)
endlocal
QPROCESS command is not so powerful as TASKLIST and is limited in showing only 12 symbols of process name but should be taken into consideration if TASKLIST is not available.
More simple usage where it uses the name if the process as an argument (the .exe suffix is mandatory in this case where you pass the executable name):
@echo off
:check_process
setlocal
if "%~1" equ "" echo pass the process name as forst argument && exit /b 1
:: first argument is the process you want to check if running
:: .exe suffix is mandatory
set "process_to_check=%~1"
QPROCESS "%process_to_check%" >nul 2>&1 && (
    echo process %process_to_check%  is running
) || (
    echo process %process_to_check%  is not running
)
endlocal
The difference between two ways of QPROCESS usage is that the QPROCESS * will list all processes while QPROCESS some.exe will filter only the processes for the current user.
Using WMI objects through windows script host exe instead of WMIC is also an option.It should on run also on every windows machine (excluding the ones where the WSH is turned off but this is a rare case).Here bat file that lists all processes through WMI classes and can be used instead of QPROCESS in the script above (it is a jscript/bat hybrid and should be saved as .bat):
@if (@X)==(@Y) @end /* JSCRIPT COMMENT **
@echo off
cscript //E:JScript //nologo "%~f0"
exit /b
************** end of JSCRIPT COMMENT **/
var winmgmts = GetObject("winmgmts:\\\\.\\root\\cimv2");
var colProcess = winmgmts.ExecQuery("Select * from Win32_Process");
var processes =  new Enumerator(colProcess);
for (;!processes.atEnd();processes.moveNext()) {
    var process=processes.item();
    WScript.Echo( process.processID + "   " + process.Name );
}
And a modification that will check if a process is running:
@if (@X)==(@Y) @end /* JSCRIPT COMMENT **
@echo off
if "%~1" equ "" echo pass the process name as forst argument && exit /b 1
:: first argument is the process you want to check if running
set process_to_check=%~1
cscript //E:JScript //nologo "%~f0" | find /i "%process_to_check%" >nul 2>&1 && (
    echo process %process_to_check%  is running
) || (
    echo process %process_to_check%  is not running
)
exit /b
************** end of JSCRIPT COMMENT **/
var winmgmts = GetObject("winmgmts:\\\\.\\root\\cimv2");
var colProcess = winmgmts.ExecQuery("Select * from Win32_Process");
var processes =  new Enumerator(colProcess);
for (;!processes.atEnd();processes.moveNext()) {
    var process=processes.item();
    WScript.Echo( process.processID + "   " + process.Name );
}
The two options could be used on machines that have no TASKLIST. 
The ultimate technique is using MSHTA . This will run on every windows machine from XP and above and does not depend on windows script host settings. the call of MSHTA could reduce a little bit the performance though (again should be saved as bat):
@if (@X)==(@Y) @end /* JSCRIPT COMMENT **
@echo off
setlocal
if "%~1" equ "" echo pass the process name as forst argument && exit /b 1
:: first argument is the process you want to check if running
set process_to_check=%~1
mshta "about:<script language='javascript' src='file://%~dpnxf0'></script>" | find /i "%process_to_check%" >nul 2>&1 && (
    echo process %process_to_check%  is running
) || (
    echo process %process_to_check%  is not running
)
endlocal
exit /b
************** end of JSCRIPT COMMENT **/
   var fso= new ActiveXObject('Scripting.FileSystemObject').GetStandardStream(1);
   var winmgmts = GetObject("winmgmts:\\\\.\\root\\cimv2");
   var colProcess = winmgmts.ExecQuery("Select * from Win32_Process");
   var processes =  new Enumerator(colProcess);
   for (;!processes.atEnd();processes.moveNext()) {
    var process=processes.item();
    fso.Write( process.processID + "   " + process.Name + "\n");
   }
   close();