VBA, 60 59 Bytes

i=30:For l=1To 30:?String(i," ")&"+":i=i+Sgn(Rnd()-.5):Next

Paste it in the Immediate window and hit enter. (Make sure explicit declaration is turned off!)

Far more likely to move than to stay in a line (i.e. actions are not equally weighted) but that was not a specified requirement (Fortunately!)

{EDIT} Saved 1 byte by removing the space between =1 and To

VBA, 60 59 49 Bytes

For l=1To 30:?Spc(30+i)"+":i=i+Sgn(Rnd()-.5):Next

Paste it in the Immediate window and hit enter. (Make sure explicit declaration is turned off!)

Far more likely to move than to stay in a line (i.e. actions are not equally weighted) but that was not a specified requirement (Fortunately!)

{EDIT} Saved 1 byte by removing the space between =1 and To

{EDIT2} Saved 10 bytes thanks to remoel's comment

Old Versions:

'V1
i=30:For l=1 To 30:?String(i," ")&"+":i=i+Sgn(Rnd()-.5):Next
'V2
i=30:For l=1To 30:?String(i," ")&"+":i=i+Sgn(Rnd()-.5):Next

VBA, 60 Bytes

i=30:For l=1 To 30:?String(i," ")&"+":i=i+Sgn(Rnd()-.5):Next

Paste it in the Immediate window and hit enter. (Make sure explicit declaration is turned off!)

Far more likely to move than to stay in a line (i.e. actions are not equally weighted) but that was not a specified requirement (Fortunately!)

VBA, 60 59 Bytes

i=30:For l=1To 30:?String(i," ")&"+":i=i+Sgn(Rnd()-.5):Next

Paste it in the Immediate window and hit enter. (Make sure explicit declaration is turned off!)

Far more likely to move than to stay in a line (i.e. actions are not equally weighted) but that was not a specified requirement (Fortunately!)

{EDIT} Saved 1 byte by removing the space between =1 and To